numerical modelling of the gas dynamics of a prototype free-piston engine · pdf...

Numerical Modelling of the Gas

Dynamics of a Prototype Free-Piston

Engine

Gregory Paul Gibbes

Submitted in fulfilment of the degree of

Doctor of Philosophy

University of Technology, Sydney

Australia

2011

keywords: free-piston, 1D gas dynamics, two-stroke, specific heat, chemical

equilibrium, charging, scavenging, tuning

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CERTIFICATE OF AUTHORSHIP/ORIGINALITY

I certify that the work in this thesis has not previously been submitted for a degree

nor has it been submitted as part of requirements for a degree except as fully

acknowledged within the text.

I also certify that the thesis has been written by me. Any help that I have received in

my research work and the preparation of the thesis itself has been acknowledged. In

addition, I certify that all information sources and literature used are indicated in the

thesis.

……………………………..

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ACKNOWLEDGEMENTS

I would first like to acknowledge the men who created Pempek Engine project.

Edward Wechner, who designed most of engine’s hardware, and Bert Van Der

Broek for his company’s provision of the Pempek scholarship and for his leadership.

Thanks also goes to Douglas Carter who designed the electrical systems, and who

was a constant, friendly, professional source of ideas and fruitful discussion. Thanks

to Slobadan Ilic, who was likewise a friendly and helpful member of the small

Pempek engine team, and who worked tirelessly to manufacture and assemble much

of the engine. I was glad for his encouragement, advice, and readiness to share

information.

My supevisor Guang Hong has been supportive over this long road, and has

provided much guidance, insight and advice which, I hope, has borne fruit in making

this thesis more useful to others. Thank you for your hard work to support me, even

when we didn’t always agree. I am in your debt.

Thanks also to Phyllis Agius for her ready helpfulness in all postgraduate student

matters, both to me and numerous other students.

Thanks to Matt Gaston, Peter Brady and John Reizes who were always generous

with their time, as I learned the art of CFD modelling early in this project.

Thanks go to the many people who I have learned from but never met except through

the pages of technical papers and text books. In particular, I would like to

acknowledge the late Gordon P. Blair, whose method of modelling 1D gas dynamics

I have adapted for this work. Thanks to Samuel J. Kirkpatrick who’s carefully

published experimental work was invaluable to me in validating my code. Thanks

also to John D. Anderson Jr. from whom I learned much about compressible flow;

and Markus Klein and Gary L. Borman in the field combustion modelling.

Thanks to Randy Lewis, a kindred spirit in the joys 1D gas dynamic modelling, who

provided valuable feedback for the material which appears in Chapter 4.

Thanks to my research friends here at UTS from all corners of the globe: Janitha

Wijesinghe, Reza Fathollahzadeh, Fabio Cumbe, Wade Smith, Ulrike Dackermann,

Debbie Marsh, Fook Choi, Minh Nguyen, Kifayah Amar, Dang Ho, Thanh Nguyen,

Jiping Niu, Wen Xing, Zhinous Zabihi, Xiaohang Pang, Ibrahim El-Saliby and

Sherub Phuntsho.

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. . . .

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ABSTRACT

Free-piston internal combustion engines found commercial success as air

compressors in the 1920’s and 1930’s, and afterward as gas turbine gasifiers for

stationary applications. Since that time they have failed to see commercial

application, however in the last decade or so there has been a resurgence of interest

in free-piston engines because of their ostensible simplicity and in the flexibility

afforded by an unconstrained piston.

This thesis reports the testing and modelling on a free-piston engine by Pempek

Systems Pty. Ltd. It is an opposed cylinder, electric machine, operating on a two

stroke cycle with direct fuel injection. Analysis of experimental cylinder pressure

shows that while compression ignition is suitably fast and reliable, the Pempek

engine suffers from (among other things) low charging efficiency. The aim of the

modelling work is to understand the reasons for this, and to investigate design

options for improvement.

A comprehensive, generally applicable 1D gas dynamics engine model has been

developed. The important features of this model are described in some detail. While

the model builds on existing methods, a number of unique contributions have been

made. A chemical equilibrium code was developed which is computationally

efficient and flexible. The 1D gas dynamics method is based on a method developed

at Queens University, Belfast (QUB) in the early 1990’s but has been thoroughly re-

worked in the way it handles friction, gas property changes and heat transfer. The

originally first order accurate method has been changed to second order, and a way

of preserving full mass conservation has been developed. An unsteady heat transfer

model is proposed. A comprehensive boundary solution is presented, which has

relevance to all 1D gas dynamics models. The gas dynamics model is validated

against extensive single shot data from QUB, and also against some experimental

engine-run data.

The 1D gas dynamics engine model is used to assess the viability of utilising exhaust

pipe tuning to drive the charging process of the Pempek engine. Simulation results

show that it is possible to charge the engine using exhaust gas dynamics alone.

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ix

PREFACE

At a stand at the World Energy Congress 2004 in Sydney I met Bert Van Der Broek

and Edward Wechner and was introduced to the Pempek free-piston engine. Ed

explained to me the fascinating features of his new engine design and invited me to

visit the workshop to see some prototype testing. This was the start of my

association with Pempek Systems, which lead to a final year project exploring the

scavenging of the engine, then on to this PhD in a similar vein.

Pempek Systems had done what few research groups had yet achieved – they had

built and run a full scale electric free-piston engine, demonstrating unequivocally

that generator based piston motion control was accurate and robust. Their working

prototype was an excellent platform from which to launch a theoretical investigation

- which was aimed at providing tools to interpret results, and doing predictive

modelling to guide future directions of the project.

A little should be said about the contents of this thesis which follow from the

requirements of the project. There are two main topics addressed. The first is the

developing technology of free-piston engines. This is the subject of the first two

chapters. The second topic is that of engine modelling and makes up the middle part

of the thesis. Even though the modelling was developed for the Pempek project, it is

nonetheless broadly applicable to all IC engines and even to other fields. Thus, the

sections on modelling can be read profitably without concern for the preceding

chapters on free-piston engine technology. Likewise, readers with little interest in

physics and methods of modelling will be able to read the sections reporting free-

piston engine technology. The final section of the thesis takes the engine model and

looks at two possible design variations for the Pempek engine.

- Synopsis -

Chapter 1 - Free-piston Engines – overview of developments

Surveys the current state of the art in free-piston engine technology. The survey

shows that despite the relative immaturity of the field, promising solutions have been

found for the main difficulties, such as piston motion control.

Chapter 2 - Pempek free-piston engine – details and experimental results

Describes the Pempek free-piston engine project in some detail, highlighting the

successful piston motion control, and describing some of the difficulties that were

faced, in particular low combustion energy and high compressor power

Preface x

consumption. In order to explore the potential for lower compressor pressure, it was

deemed necessary to analyse the gas dynamics of inlet and exhaust systems. This is

the motivation for the modelling work which follows.

Chapter 3 - Thermodynamic and gas property models

Describes three key components of the engine model – namely the single zone

thermodynamic cylinder model, the gas property model and the chemical

equilibrium model.

Chapter 4 - Unsteady 1D gas dynamics model

Describes the unsteady gas dynamics model, which was based on an existing

method, but with several modifications. Concludes with some simple validation

cases.

Chapter 5 - Other sub models

Describes miscellaneous other parts of the engine model which were not covered in

the previous two chapters.

Chapter 6 - The engine model – integrating all sub models

Explains the integration of all the sub models into the overall engine model

Chapter 7 - Validations using experimental results

Validates the gas dynamics model against a suit of single shot experiments, and also

a superficial comparison to measured data from the Pempek engine.

Chapter 8 - Predictive modelling

Applies the gas dynamics model to the original Pempek engine but with a modified

low pressure compressor, and a tuned exhaust pipe. Simulation results show that

low compressor pressure operation is possible. Next, a radical design modification

is proposed, and the gas dynamics model is used to test the viability of un-boosted

charging. These two applications of the gas dynamics model demonstrate the

usefulness of the model, and the sort of design options that are available for free-

piston engines to take advantage of gas dynamics to improve and control charging.

Chapter 9 - Summary and conclusion

Summarises the specific findings for the Pempek project, summarises the model

scope and usefulness, and lists the unique contributions of the thesis. A list of

suggested further work is also included.

Preface xi

Appendices

A wide range of material with further details of free-piston engine projects and the

engine model.

Appendix I Review of recent free-piston engine projects

Appendix II Specific heats of a reacting mixture

Appendix III Further applications of the energy equation

Appendix IV Tables of Thermodynamic Properties

Appendix V Method for Calculating Chemical Equilibrium of Combustion Products

Appendix VI Derivation of fundamental one dimensional unsteady gas equation

Appendix VII Derivation of Boundary Flow Equations

Appendix VIII Model Data Structures

Appendix IX 2nd Order Interpolation – further details

Appendix X Re-meshing Criteria and Method

Appendix XI Single Shot Experiments Cross Reference

Appendix XII Derivation of normal shock equations

Appendix XIII Rayleigh and Fanno Flow

Appendix XIV Graphical user interface screen shots

Appendix XV Table of contents of data CD

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xiii

TABLE OF CONTENTS

Abstract ................................................................................................................................. vii

Preface .................................................................................................................................... ix

Table of Contents................................................................................................................. xiii

List of Figures ..................................................................................................................... xvii

List of Tables ..................................................................................................................... xxvii

Nomenclature ..................................................................................................................... xxix

Acronyms ......................................................................................................................... xxxiii

Chapter 1 Free-piston Engines – overview of developments .......................................... 1

1.1 Introduction ............................................................................................................. 2

1.2 Summary of free-piston projects ............................................................................. 4

1.3 Piston Motion Control ............................................................................................. 6

1.4 Discussion on free-piston engines state of the art ................................................... 9

1.5 Free-piston Engine Modelling .............................................................................. 11

Chapter 2 Pempek free-piston engine – details and experimental results ................... 19

2.1 Overview of the project......................................................................................... 19

2.2 Details of the engine ............................................................................................. 21

2.3 Cylinder pressure analysis .................................................................................... 27

2.4 Compressor analysis ............................................................................................. 31

2.5 Summary ............................................................................................................... 33

Chapter 3 Thermodynamic and gas property models ................................................... 37

3.1 Thermodynamic control volume model ................................................................ 38

3.2 Gas mixture property model ................................................................................. 45

3.3 Reacting gas mixture model .................................................................................. 49

Chapter 4 Unsteady 1D gas dynamics model ................................................................. 55

4.1 Introduction ........................................................................................................... 55

4.2 Theoretical Basis ................................................................................................... 61

4.3 Wave Propagation ................................................................................................. 63

4.4 Flow Boundary solution ........................................................................................ 68

Table of Contents xiv

4.5 Mass and thermal energy transport ....................................................................... 79

4.6 Validation using analytical results ........................................................................ 83

4.7 Summary ............................................................................................................... 89

Chapter 5 Other sub models ............................................................................................ 91

5.1 Duct friction and heat transfer ............................................................................... 92

5.2 Combustion cylinder models ............................................................................... 105

5.3 Separated flow model .......................................................................................... 112

5.4 Flow area coefficient maps ................................................................................. 115

5.5 Multi body dynamics........................................................................................... 117

Chapter 6 The engine model – integrating all sub models .......................................... 123

6.1 Overview of model building blocks .................................................................... 124

6.2 Calculation sequence for a complete time-step evaluation ................................. 125

6.3 Programing details of the engine model .............................................................. 127

Chapter 7 Validations using experimental results ....................................................... 129

7.1 Description of the single shot tests ...................................................................... 130

7.2 Slide valve tests ................................................................................................... 134

7.3 P1 driven simulation ........................................................................................... 141

7.4 Straight pipe shots ............................................................................................... 142

7.5 Converging flow ................................................................................................. 147

7.6 Diverging flow .................................................................................................... 150

7.7 Modelling the Pempek engine ............................................................................. 161

Chapter 8 Predictive modelling ..................................................................................... 171

8.1 Optimising the existing layout ............................................................................ 172

8.2 Port admission layout .......................................................................................... 175

8.3 Gas dynamics driven scavenging ........................................................................ 178

8.4 Conclusion .......................................................................................................... 183

Chapter 9 Summary and conclusion ............................................................................. 185

9.1 Findings for the Pempek project ......................................................................... 186

9.2 The model, its scope and usefulness ................................................................... 187

9.3 Unique contributions to modelling art ................................................................ 188

9.4 Contribution to free-piston engine research ........................................................ 190

9.5 Further Work ....................................................................................................... 191

Table of Contents xv

Publications ......................................................................................................................... 193

References ........................................................................................................................... 195

Appendices .......................................................................................................................... 207

Appendix I Review of recent free-piston engine projects ................................................ 209

Appendix II Specific heats of a reacting mixture ............................................................. 235

Appendix III Further applications of the energy equation ............................................... 237

Appendix IV Tables of Thermodynamic Properties ........................................................ 241

Appendix V Method for Calculating Chemical Equilibrium of Combustion Products ... 247

Appendix VI Derivation of fundamental one dimensional unsteady gas equation .......... 255

Appendix VII Derivation of Boundary Flow Equations .................................................. 259

Appendix VIII Model Data Structures ............................................................................. 263

Appendix IX 2nd Order Interpolation – further details ..................................................... 269

Appendix X Re-meshing Criteria and Method ................................................................ 271

Appendix XI Single Shot Experiments Cross Reference ................................................. 273

Appendix XII Derivation of normal shock equations ...................................................... 275

Appendix XIII Rayleigh and Fanno Flow ........................................................................ 281

Appendix XIV Graphical user interface screen shots ...................................................... 285

Appendix XV Table of contents of data CD .................................................................... 291

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xvii

LIST OF FIGURES

Figure 1-1 Various free-piston engine layouts ............................................................ 2

Figure 1-2 Two control methodologies ....................................................................... 8

Figure 1-3 Scavenging modes analysed by Goldsborough [58] ................................ 15

Figure 1-4 Simulated 1D model compared to experiment for pressure in exhaust

pipe. Larmi et al [74] ....................................................................................... 17

Figure 2-1 Spark ignition prototype .......................................................................... 20

Figure 2-2 Cross section of engine module (shown vertically) ................................. 22

Figure 2-3 Typical exhaust valve actuator trajectory ................................................ 23

Figure 2-4 Real vs. target mover velocity ................................................................. 25

Figure 2-5 Example engine run indicated work (efficiency indicated by black

markers) ............................................................................................................ 26

Figure 2-6 Sample indicator plot – fired cycle .......................................................... 27

Figure 2-7 Sample indicator plot - motored cycle ..................................................... 27

Figure 2-8 Apparent heat release............................................................................... 28

Figure 2-9 Cylinder and compressor pressure during scavenging ............................ 29

Figure 2-10 Comparison of cylinder pressure with and without exhaust pipe .......... 30

Figure 2-11 Indicator plot of compressor .................................................................. 31

Figure 2-12 Proposed advanced passive inlet valve design [96] ............................... 32

Figure 3-1 Numerical solution of the energy equation .............................................. 40

Figure 3-2 Pressure history and error for various time steps ..................................... 42

Figure 3-3 Prescribed fuel burn rate and cylinder volume ........................................ 43

Figure 3-4 Pressure and temperature for combustion case ........................................ 43

Figure 3-5 Error for various time steps ..................................................................... 44

Figure 3-6 Specific heat Cp of common exhaust gas species .................................... 45

Figure 3-7 Typical pressure error incurred for setting =1.4 .................................... 45

Figure 3-8 Variation of with temperature and equivalence ratio for unburned and

burned mixture ................................................................................................. 47

List of Figures xviii

Figure 3-9 Equilibrium species mass fractions of a fuel-air mixture at various

temperatures and pressure ................................................................................ 52

Figure 3-10 Equilibrium species mass fractions of a fuel air mixture with varying

fuel to oxygen ratio ........................................................................................... 52

Figure 4-1 Evolving mass flow rate into an idealised duct ....................................... 55

Figure 4-2 A right travelling pressure wave .............................................................. 61

Figure 4-3 Oppositely moving pressure waves ......................................................... 62

Figure 4-4 Advancing pressure waves by one time step ........................................... 63

Figure 4-5 Second order interpolation of pressure waves ......................................... 64

Figure 4-6 Modifying pressure waves to account for heat transfer and mass

conservation ...................................................................................................... 65

Figure 4-7 Re-meshing a duct ................................................................................... 67

Figure 4-8 Detection of a travelling shock ................................................................ 67

Figure 4-9 Duct cell boundary nodes in space and time ........................................... 68

Figure 4-10 Variation of gas properties around a node in space and time ................ 68

Figure 4-11 Catalogue of all flow types considered .................................................. 70

Figure 4-12 A typical flow boundary showing all flow properties ........................... 72

Figure 4-13 Mass and thermal transport.................................................................... 79

Figure 4-14 Calculating boundary flow properties ................................................... 81

Figure 4-15 Temperature transport with different mixing coefficients ..................... 82

Figure 4-16 Standard shock tube results ................................................................... 84

Figure 4-17 Shock tube with mass conservation ....................................................... 85

Figure 4-18 Shock tube comparison between first and second order wave

interpolation ...................................................................................................... 86

Figure 4-19 Fanno flow ............................................................................................. 87

Figure 4-20 Rayleigh flow ........................................................................................ 87

Figure 4-21 Smearing of a triangular pulse traversing 100 mesh spaces .................. 88

Figure 5-1 Coefficient of friction for flow over a flat plate [90] ............................... 92

List of Figures xix

Figure 5-2 Fluid element experiencing friction ......................................................... 93

Figure 5-3 Fluid element experiencing heat transfer ................................................. 95

Figure 5-4 Heat transfer model compared to Dittus-Boelter for steady flow ............ 99

Figure 5-5 Heat transfer model turbulent kinetic energy for different turbulence

length scales ...................................................................................................... 99

Figure 5-6 Heat transfer model for a turbulence generating inlet ........................... 100

Figure 5-7 Heat transfer model compared to Dittus-Boelter for low speed flow .... 100

Figure 5-8 Heat transfer model for different cell spacing ....................................... 101

Figure 5-9 Heat transfer modelled for single shot using different turbulence length

scales............................................................................................................... 102

Figure 5-10 Heat transfer modelled for single shot using different timestep size ... 103

Figure 5-11 Sketch of piston-cylinder crevice ........................................................ 106

Figure 5-12 Blowby CFD model mesh ................................................................... 107

Figure 5-13 Blowby correlation .............................................................................. 108

Figure 5-14 Finding fuel injection enthalpy ............................................................ 109

Figure 5-15 Spark ignition combustion rate model ................................................. 110

Figure 5-16 Compression ignition combustion rate model ..................................... 111

Figure 5-17 Control volume for applying the momentum equation to a diffusing flow

........................................................................................................................ 112

Figure 5-18 Modified area ratio for tapered ducts................................................... 114

Figure 5-19 Example flow area coefficient map ..................................................... 116

Figure 5-20 Cutaway of FP3 showing moving parts ............................................... 117

Figure 5-21 Forces on the mover ............................................................................ 117

Figure 5-22 Forces on the exhaust valves ............................................................... 118

Figure 5-23 Aero force coefficients for normal flow through exhaust and inlet valves

........................................................................................................................ 119

Figure 5-24 Forces on the passive inlet valves ........................................................ 120

Figure 5-25 Typical collision trajectory .................................................................. 122

List of Figures xx

Figure 6-1 Integration of sub models to make an engine model ............................. 123

Figure 7-1 Straight pipe........................................................................................... 130

Figure 7-2 Straight pipe with density discontinuity ................................................ 131

Figure 7-3 Sudden contraction ................................................................................ 131

Figure 7-4 Convergent taper ................................................................................... 131

Figure 7-5 Sudden enlargement .............................................................................. 132

Figure 7-6 Divergent taper ...................................................................................... 132

Figure 7-7 Short megaphone ................................................................................... 132

Figure 7-8 Long megaphone ................................................................................... 132

Figure 7-9 Flow area coefficients used for slide valve ........................................... 135

Figure 7-10 Slide valve Prel =1.5 bar, Trel=293K ..................................................... 136

Figure 7-11 Slide valve Prel =1.5 bar, Trel=293K, air in cylinder, CO2 in pipe ....... 136

Figure 7-12 Slide valve Prel =1.5 bar, Trel=293K, CO2 in cylinder, air in pipe........ 136


Figure 7-14 Slide valve Prel =2.4 bar, Trel=293K, air in cylinder, CO2 in pipe........ 137

Figure 7-15 Slide valve Prel =2.4bar, Trel=293K, CO2 in cylinder, air in pipe......... 137

Figure 7-16 Slide valve Prel =2 bar, Trel=623K ........................................................ 138

Figure 7-17 Slide valve Prel =5 bar, Trel=623K ........................................................ 138



Figure 7-20 Slide valve Prel =0.5 bar, Trel=293K short pipe shot ............................ 139



Figure 7-23 Straight pipe P2, Prel =0.5 bar, Trel=293K ............................................ 143




List of Figures xxi

Figure 7-27 Density discontinuity P3, CCC, Prel =2.4 bar, Trel=293K, closed end . 144

Figure 7-28 Density discontinuity P1, AAC, Prel =1.5 bar, Trel=293K .................... 144




Figure 7-32 Density discontinuity P1, CCA, Prel =1.5 bar, Trel=293K .................... 145




Figure 7-36 Flow area coefficients used for sudden area change ........................... 147

Figure 7-37 Sudden contraction 53mm P1, Prel =2.4bar, Trel=293K ....................... 148

Figure 7-38 Convergent taper 53mm P1, Prel =2.4bar, Trel=293K........................... 148

Figure 7-39 Sudden contraction 53mm, P3, Prel =2.4bar, Trel=293K ...................... 148

Figure 7-40 Convergent taper 53mm, P3, Prel =2.4bar, Trel=293K .......................... 148

Figure 7-41 Sudden contraction 80.2mm, P1, Prel =2.4bar, Trel=293K ................... 149

Figure 7-42 Convergent taper 80.2mm, P1, Prel =2.4bar, Trel=293K ....................... 149

Figure 7-43 Sudden contraction 80.2mm, P3, Prel =2.4bar, Trel=293K ................... 149

Figure 7-44 Convergent taper 80.2mm, P3, Prel =2.4bar, Trel=293K ....................... 149

Figure 7-45 Sudden enlargement 53mm, P1, Prel =1.5bar, Trel=293K .................... 152

Figure 7-46 Divergent taper 53mm, P1, Prel =1.5bar, Trel=293K ............................ 152

Figure 7-47 Sudden enlargement/ Divergent taper 53mm, P3, Prel =1.5bar,

Trel=293K ........................................................................................................ 152

Figure 7-48 Sudden enlargement 53mm, P1, Prel =2.4bar, Trel=293K .................... 153

Figure 7-49 Divergent taper 53mm, P1, Prel =2.4bar, Trel=293K ............................ 153

Figure 7-50 Sudden enlargement/ Divergent taper 53mm, P3, Prel =2.4bar,

Trel=293K ........................................................................................................ 153

Figure 7-51 Sudden enlargement 80.2mm, P1, Prel =1.5bar, Trel=293K ................. 154

Figure 7-52 Divergent taper 80.2mm, P1, Prel =1.5bar, Trel=293K ......................... 154

List of Figures xxii

Figure 7-53 Sudden enlargement/ Divergent taper 80.2mm, P3, Prel =1.5bar,

Trel=293K ........................................................................................................ 154

Figure 7-54 Sudden enlargement 80.2mm, P1, Prel =2.4bar, Trel=293K ................. 155


Figure 7-56 Sudden enlargement/ Divergent taper 80.2mm, P3, Prel =2.4bar,

Trel=293K ........................................................................................................ 155


Figure 7-58 Divergent taper 105.6mm, P1, Prel =2.4bar, Trel=293K ....................... 156



Figure 7-61 Short Megaphone P1, Prel =2.0bar, Trel=293K ..................................... 157

Figure 7-62 Long Megaphone P1, Prel =2.0bar, Trel=293K ..................................... 157


Figure 7-64 Convergent taper 53mm, P1, Prel =0.5bar, Trel=293K.......................... 158

Figure 7-65 Sudden contraction / Convergent taper 53mm, P3, Prel =0.5bar,

Trel=293K ........................................................................................................ 158


Figure 7-67 Convergent taper 53mm, P1, Prel =0.8bar, Trel=293K.......................... 159

Figure 7-68 Sudden contraction / Convergent taper 53mm, P3, Prel =0.8bar,

Trel=293K ........................................................................................................ 159

Figure 7-69 Short Megaphone P1, Prel =2.0bar, Trel=293K, various simulation

timesteps ......................................................................................................... 160

Figure 7-70 Long Megaphone P1, Prel =2.0bar, Trel=293K, various simulation

timesteps ......................................................................................................... 160

Figure 7-71 Pempek engine inlet side ducting half section 3D view ...................... 161

Figure 7-72 Pempek engine1D model layout .......................................................... 162

Figure 7-73 Motored engine comparison with experiment ..................................... 163

Figure 7-74 Motored engine, indicator diagram ..................................................... 164

Figure 7-75 Motored engine, indicator diagram ..................................................... 164

List of Figures xxiii

Figure 7-76 Fired engine comparison with experiment .......................................... 165

Figure 7-77 Fired engine, indicator diagram ........................................................... 165

Figure 7-78 Fired engine, indicator diagram ........................................................... 166

Figure 7-79 Valve trajectories and mass flows ....................................................... 168

Figure 7-80 Cylinder mass and inlet mass .............................................................. 168

Figure 7-81 Cylinder species mass fractions ........................................................... 169

Figure 7-82 Cylinder temperature and specific heat ratio ....................................... 169

Figure 8-1 Tuned exhaust pipe layout for existing engine ...................................... 172

Figure 8-2 Results for optimised layout .................................................................. 173

Figure 8-3 Valve trajectories and mass flows (full power) ..................................... 173

Figure 8-4 Cylinder mass and inlet mass (full power) ............................................ 174

Figure 8-5 Cylinder species mass fractions (full power)......................................... 174

Figure 8-6 Modification of Pempek engine for port admission .............................. 175

Figure 8-7 Contactless pistons with air bearings ..................................................... 177

Figure 8-8 Port admission engine model layout ...................................................... 178

Figure 8-9 General results for port admission layout .............................................. 179

Figure 8-10 Port openings and mass flows (full power) ......................................... 180

Figure 8-11 Cylinder pressure during scavenging (full power) .............................. 180

Figure 8-12 Modelled inlet and exhaust plenum pressure (full power) .................. 180

Figure 8-13 Cylinder mass and inlet mass (full power) .......................................... 181

Figure 8-14 Cycle temperatures .............................................................................. 181

Figure 8-15 Cycle pressures .................................................................................... 182

Figure 8-16 Cycle pressures during scavenging ...................................................... 182

Figure I-1 Junkers four-stage free-piston air compressor [8] .................................. 210

Figure I-2 Partial cut-away diagram of the SIGMA Type GS-34 Free-Piston Gasifier

[8] ................................................................................................................... 211

Figure I-3 INNAS “Chiron” hydraulic free-piston engine [7] ................................ 212

List of Figures xxiv

Figure I-4 Tampere University of Technology hydraulic free-piston engine prototype

“Emma2” [113] .............................................................................................. 213

Figure I-5 Third Generation prototype engine [114] ............................................... 214

Figure I-6 Cutaway view of the Toyohashi university of Technology hydraulic free-

piston engine [63] ........................................................................................... 215

Figure I-7 EPA six-cylinder four-stroke FPE [32] .................................................. 216

Figure I-8 West Virginia University, second generation linear engine prototype (2-

stroke) [115] ................................................................................................... 218

Figure I-9 West Virginia University, four stroke concept [97] ............................... 219

Figure I-10 First Sandia Free-piston Linear Alternator concept [116].................... 220

Figure I-11 Sandia opposed piston layout [119] ..................................................... 221

Figure I-12 Sandia bounce chamber detail showing compressed air injection valves

on the left and vent ports on the right [119] ................................................... 222

Figure I-13 Sandia Opposed Piston Free-piston Engine [120] ................................ 223

Figure I-14 Timeline of Sandia free-piston engine project [119]............................ 223

Figure I-15 FPEC prototype electric generator – opposed cylinder, two-stroke diesel

with pnumatic exhaust valves [78] ................................................................. 224

Figure I-16 Korea Institute of Energy opposed cylinder engine [125] ................... 225

Figure I-17 Four-stroke free-piston engine concept by Nanjing University of

Science and Technology [128] ....................................................................... 227

Figure I-18 German Aerospace Center free-piston prototype on test bench [60] ... 227

Figure I-19 Linear alternator built at WVU [34] ..................................................... 228

Figure I-20 Sandia linear alternator design [117] ................................................... 229

Figure I-21 Magnaquench linear alternator stator [117] ......................................... 229

Figure I-22 University of Sheffield FPEC linear permanent magnet generator [122]

........................................................................................................................ 230

Figure I-23 Royal Institute of Technology Stockholm, Sweden transverse flux

permanent magnet generator [40] ................................................................... 230

Figure I-24 German Aerospace Center linear generator on the test stand [60] ....... 231

List of Figures xxv

Figure I-25 Design concept for a miniature HCCI free-piston engine [112] .......... 232

Figure I-26 Photo of Kvaerner test cylinder unit .................................................... 233

Figure I-27 Liquid piston air compressor prototype (high inertance model)

Vanderbilt University [73].............................................................................. 234

Figure III-1 Variation of R with temperature for products of combustion in chemical

equilibrium ..................................................................................................... 240

Figure V-1 Heat release of n-octane at various equivalence ratios and temperatures

........................................................................................................................ 253

Figure VI-1 A fluid element influenced by a pressure wave ................................... 255

Figure VII-1 Schematic of a duct boundary ............................................................ 259

Figure IX-1 Interpolating discontinuous pressure waves ........................................ 269

Figure IX-2 Handling the duct ends ........................................................................ 270

Figure IX-3 Single cell ducts .................................................................................. 270

Figure XII-1 Schematic of a normal shock ............................................................. 275

Figure XII-2 Travelling shock ................................................................................. 280

Figure XIV-1 Main screen ...................................................................................... 286

Figure XIV-2 Edit Ducts screen .............................................................................. 286

Figure XIV-3 Edit Volumes screen ......................................................................... 287

Figure XIV-4 Edit Bodies screen ............................................................................ 287

Figure XIV-5 Edit Area Coefficients screen ........................................................... 288

Figure XIV-6 Edit Functions screen ....................................................................... 288

Figure XIV-7 Create Animated plot screen ............................................................ 289

Figure XIV-8 Example Animation screen grab (shock tube problem) ................... 289

Figure XIV-9 History plot screen ............................................................................ 290

xxvi

. . . .

xxvii

LIST OF TABLES

Table 3-1 Thermodynamic cylinder model calculation sequence ............................. 41

Table 3-2 List of species considered in equilibrium calculation ............................... 50

Table 3-3 Equilibrium equations ............................................................................... 51

Table 4-1 Shock tube setup ....................................................................................... 83

Table IV-1 Enthalpy of some combustion products ................................................ 241

Table IV-2 Specific heat of some combustion products ......................................... 242

Table IV-3 Properties of some fuels (various sources [30, 62, 107]) ...................... 243

Table IV-4 Equilibrium equations ........................................................................... 244

Table IV-5 Equilibrium constants for selected reactions ........................................ 245

Table XIV-1 Cross reference figure numbers for single shot data .......................... 273

xxviii

. . . .

xxix

NOMENCLATURE

Symbols

A area (m2)

a speed of sound (m/s)

a0 isentropic reference speed of sound (m/s)

a acceleration (m/s2)

moles of atomic element i.e. (mol)

circumference or wetted perimeter of a duct (m)

Coefficient of friction (Fanning friction factor) (-)

Coefficient of heat transfer (W/m2/K)

specific heat at constant pressure and constant volume (J/kg/K)

specific heat at constant pressure and constant volume (J/mol/K)

frozen specific heats (see Appendix II) (J/kg/K)

c wave velocity (m/s)

hydraulic diameter (m)

F force (N)

Enthalpy, enthalpy flow rate (J, J/s)

specific enthalpy, enthalpy of formation (J, J/kg, J/mol)

thermal conductivity (W/m/K)

Length

Large eddy length

M momentum (kg.m/s)

M molar mass (g/mol)

mass, mass flow rate (kg, kg/s)

Total mixture moles (mol)

Species moles (mol)

Nomenclature xxx

P, P0 pressure, reference pressure (absolute pressure, Pa)

heat transfer rate (J/s, J/kg/s)

gas constant (J/kg/K)

universal gas constant (8.314472 J/mol/K)

Reynolds Number, turbulent Reynolds number (-)

T temperature (K)

isentropic reference temperature (K)

t time (s)

u, u0 fluid velocity, quiescent fluid velocity (m/s)

turbulence intensity, RMS of fluctuating velocity (m/s)

internal energy, specific internal energy (J, J/kg, J/mol)

v velocity (m/s)

V, v volume, specific volume (m3, m3/kg)

work rate (J/s), specific work rate (J/kg/s),

21

0PPX pressure amplitude ratio (-)

species mole fraction (-)

ratio of specific heats (-)

ratio of frozen specific heats (-)

viscosity (N.s/m2)

Courant number (-)

density (kg/m3)

etc species molar concentration (kmol/m3)

Nomenclature xxxi

Subscripts

L leftward

R rightward

i iteration, incident

f flow

prev previous

r reflected

s species

xxxii

. . . .

xxxiii

ACRONYMS

BDC Bottom dead centre

BTDC Before top dead centre

CFD Computational Fluid Dynamics

FPEC Free-piston Energy Converter, project name for a European free-

piston engine consortium

HCCI Homogenous Charge Compression Ignition

IMEP Indicated mean effective pressure

QUB Queens University Belfast

TDC Top dead centre

WRT With respect to

xxxiv

. . . .

1

Chapter 1 Free-piston Engines – overview of

developments

Free-piston combustion engines found commercial success as air compressors in the

1920’s and 1930’s, and then as gas turbine gasifiers for stationary applications.

However, since that time they have failed to see commercial application. There is

sporadic literature in the intervening years on several renewed attempts, and in the

last decade or so there has been a resurgence of interest in the idea.

This chapter gives an outline for the state of free-piston engine development at this

time. Basic engine layout options are summarised, recent published engine projects

are listed, piston motion control is discussed, and the general state of the art is

critiqued. Next, several of the more notable modelling efforts are described, with

emphasis on modelling related to the modelling conducted here. An in-depth

description of the Pempek engine project and experimental results follows in Chapter

2.

Chapter 1 Free-piston Engines – overview of developments 2

1.1 Introduction

Free-piston engine layouts can be usefully categorised by piston-cylinder

arrangement. The three basic possibilities are shown in Figure 1-1.

Single ended layouts have only one combustion cylinder, and must have some way

of returning the piston for the next compression stroke. Single piston hydraulic

machines can have lighter movers which result in higher speed and higher power

output. For hydraulic cases, the single ended layout can allow pulse pause control.

Electric machines usually use a so-called gas spring to return the piston. The mass

of air in the bounce chamber can be varied to help control piston motion.

Figure 1-1 Various free-piston engine layouts

In the opposed cylinder arrangement, there are two identical combustion chambers

and a single moving assembly, and a centrally located pump or generator. This

arrangement has the advantage of not needing a bounce mechanism to return the

piston at BDC, since the opposite cylinder achieves this function. The engine is

symmetrical, as is the motion of the mover. This means that the trajectory at BDC

must be the same as the trajectory at TDC.

Opposed piston machines are essentially two single ended machines placed so that

their combustion pistons occupy either end of the same combustion cylinder. Thus

there are two generators/pumps and two bounce mechanisms. These machines enjoy

Opposed cylinder

Single ended

Opposed piston

generator or pump cylinder

piston

cylinder

Bounce mechanism


all of the advantages and suffer the disadvantages of single ended machines with the

additional advantage of highly effective uni-flow port scavenging being possible,

intrinsic dynamic balance, and increased effective stroke to bore ratio. At the same

time, there is the added constraint on the control system of keeping the pistons

carefully synchronised.

Other possible layouts include a four stroke engine made up of two opposed cylinder

machines with a common mover such as proposed by [97], or the six cylinder layout

described in [32]. The eight cylinder two stroke engine proposed by [33] is made of

four opposed cylinder modules arranged so that the inertia of the four synchronised

movers cancel.

Most free-piston engine concepts operate on a two stroke cycle. The two stroke

cycle relies on rapid transfer of cylinder gases around the bottom of the piston stroke

during which time both inlet and exhaust ports/valves may be open.

If the fuel is pre-mixed with air outside the cylinder, then fuel short circuiting must

be avoided by careful management of the scavenging process, such as using a

reduced scavenging ratio. This in turn means that it is impossible to obtain a

homogenous cylinder charge and several researchers have found spark ignition to be

somewhat unreliable due to high residual exhaust gas fraction. In practice

compression ignition has often been used to give reliable combustion, even when

scavenging is incomplete.

Direct fuel injection allows higher scavenging ratios to be employed, since only pure

air may short circuit the cylinder. High speed, high pressure common rail fuel

injectors which allow multiple injections during the compression and combustion

process open opportunities to create customised fuel mixture patterns.

The speed at which scavenging must be carried out generally necessitates some form

of pressurisation on the inlet air, and this is usually carried out by means of a

separate compressor, or an integrated crank-case-style compression volume.

A number of researchers have concluded that due to difficulties of the two stroke

cycle it was worth pursuing a mechanically more-challenging four stroke concept.

Eg [32, 45, 97, 128].


1.2 Summary of free-piston projects

The following brief overview of free piston engine projects is not exhaustive, and

focuses on recently reported projects.

1.2.1 Early free-piston engines

Free-piston engines are not new [8]. Prolific inventor Raúl Pateras Pescara

developed a number of designs for free-piston air compressors beginning in 1922. At

around the same time the German company Junkers began developing similar air

compressors, which were used to provide compressed air for torpedo launch tubes on

German submarines. The next major phase in free-piston technology was introduced

in 1944 with the Pescara designed 600 kW SIGMA gasifiers. In the intervening

years several attempts have been made at producing smaller gasifiers, such as the

General Motors GMR 4-4 Hyprex but without success [8].

1.2.2 Hydraulic free-piston engines

Following the gasifier type free-piston engine, hydraulic engines have been

developed to various degrees, however none have yet (to the author’s knowledge)

been produced commercially. Hydraulic free-piston engines utilise hydraulic oil to

extract piston work. The earlier success of hydraulic engines compared to electric

free-piston engines can be attributed to the ubiquitous nature of high performance

hydraulic componentry compared to the less advanced field of high performance

linear electric machines, and the relatively recent arrival of high coercivity

permanent magnets and appropriate solid state electric power technology. In

addition, some researchers suggest there is less difficulty in adequately controlling

hydraulic free-piston machines. The following overview lists hydraulic free-piston

engines published since about the year 2000 onwards.

The Chiron engine by Dutch company INNAS is a single ended, direct injection

compression ignition machine, and was an advanced prototype in 2000 [7]. There

has been little further word since that time. Tampere and Helsinki Universities of

Technology designed and built several opposed cylinder compression ignition

prototypes [114]. Toyohashi University of Technology have had a long running

project based on an opposed piston uni-flow port scavenged machine [63]. The U.S.

EPA and NVFEL engaged in an intensive free-piston engine development program

as part of their hydraulic vehicle program. Two stroke and four stroke variants were


developed, both based on an opposed cylinder, compression ignition layout. Exhaust

was via hydraulic or cam actuated poppet valves [32]. Beijing Institute of

Technology have a single ended hydraulic machine under development [126].

1.2.3 Electric free-piston engines

West Virginia University have had a long running free-piston engine project with

several prototypes reported [64, 91, 115]. All have been opposed cylinder spark

ignition machines. Sandia National Laboratories have had an ongoing free-piston

engine project which grew out of research into hydrogen and high compression ratio

engines. Initially an opposed cylinder machine was designed [116], but more

recently an opposed piston machine has been constructed with a view to testing a

range of different fuels [120]. A consortium of companies and universities obtained

European funding to develop the Free-piston Energy Converter (FPEC). It is an

opposed cylinder diesel machine with pneumatic exhaust valves [78], and is

probably the most advanced electric free-piston engine to date. The University of

Newcastle upon the Tyne began a free-piston engine project in 1999. More recently,

numerous papers have been published detailing various modelling efforts for a

proposed two stroke single ended machine [80]. Australian company Pempek

Systems developed and tested a compact two-stroke opposed cylinder machine [33].

Details of this project are given in Chapter 2. Loughborough University partnered

with Sheffield University and Lotus Engineering Ltd. in 2005 under a UK

government EPSRC grant to develop a four-stroke free-piston engine [45]. Other

electric free-piston engine projects are reported from the Korea Institute of Energy

[125], Malaysian Ministry of Science, technology and Environment [17], Shanghai

Jiao Tong University [75], Nanjing University of Science and Technology [128], and

the German Aerospace Center [60].

***

The above catalogue of recent and current free-piston engine projects demonstrates

the renewed activity in this field in the past decade. Greater detail on these and other

projects is provided in Appendix I, including an overview of notable linear generator

projects.


1.3 Piston Motion Control

Unlike a crank driven piston, a free-piston travels under the combined influence of

gas pressure and any load forces. The question arises as to whether this combination

of forces results in inherently stable or unstable reciprocations. The question of

engine control is central to recent work by Mikalsen and Roskilly [80-88] which

focuses on electric free-piston engines. Their modelling work discovered that in the

absence of suitable closed loop control input, the compression ratio from stroke to

stoke is unstable.

Intriguingly however, some practical free-piston engines have been reported to

operate stably with no closed loop control. The hydraulic free-piston engine from

Tampere University of Technology was capable of steady operation at a fixed

hydraulic load and fixed fuelling level [113], and was more stable in practice than

modelling suggested [74]. All of the prototypes reported from West Virginia

University [64, 91, 115] have been run at constant fuel level and variable load. Even

more surprisingly, the load used in these cases was a friction brake which is not a

viscous-type load such as a passive alternator would be. This author believes that

the demonstrated stability of the WVU machines is mainly due to the damping

influence of combustion chamber leakage (crevice volumes and blowby). The

stability in the Tamper machine was conditional on steady load and is likely to be the

combined result of combustion chamber losses and hydraulic losses at over-stroke

conditions.

Nevertheless, the majority of recent researchers have reported machines with

feedback control. The free-piston gasifier from Kvaerner ASA [66, 67] uses a

combination of fuel level, valve timing and bounce chamber pressure to achieve

piston control. Tikkanen and Vilenius [114] report a control system designed for

operating a refined version of the hydraulic machine that was originally run

uncontrolled. Control is affected by varying fuel mass. Information about changes

in the hydraulic load is used by the controller to pre-empt the resulting change to

piston motion and allow rapid response in changes to fuelling level. The controller

was tested on a Matlab/Simulink model of the engine and showed good ability to

control compression ratio, provided that the rate of change in load level was kept

within certain limits. No experimental results were reported. The hydraulic free-

piston engine proposed by the Beijing Institute of Technology [126] will also try to

achieve piston control by varying fuel levels on a stroke by stroke basis.


In contrast to using fuel level as a control input, the hydraulic machines developed

by the US EPA [32] achieved piston motion control by varying the hydraulic power

extraction on a stroke by stroke basis in response to changes in fuel energy.

Controllable check valves in the hydraulic circuit were utilised to provide this

control and allow arbitrary hydraulic outlet pressure. The Toyohashi University of

Technology machine [63] uses pulse-pause modulation to control power output.

Each cycle is independent, thus disturbances in one cycle are not carried over to the

next.

In the field of electric free-piston engines, the experimental free-piston engine rig

being developed at Sandia [120] uses compressed air automatically injected into the

bounce chambers to supplement combustion energy on a stroke by stroke basis. In

its present form with a fixed alternator load, this control system would have

excessive energy requirements at low load, however in the future a controllable

alternator load may alleviate this problem. The most advanced electric free-piston

engine project presently reported is the Free-piston Energy Converter (FPEC). This

machine uses a controlled electric machine allowing electromagnetic force to be

used as a control input [78]. The Pempek free-piston engine and the engine being

developed at the German Aerospace Center [60] also uses generator force as the

control input. The ambitious four stroke concept proposed at the Nanjing University

of Science and Technology [128] relies heavily on a bi-directional electric machine

to control and drive the motion of the spring mounted piston.

Mikalsen and Roskilly’s extensive numerical investigations on free-piston engine

control assume the load imposed on the mover is not a control input, though they do

acknowledge the possibility of implementing some kind of alternator control through

the use of power electronics (indeed controllability of the machine is assumed for the

purposes of starting the engine [80]). The controllability (or lack thereof) of the

mover load turns out to be a crucial differentiating factor in all of the control

strategies reported. They produce two entirely distinct control philosophies.

Figure 1-2 illustrates the two different control methods. In Figure 1-2(a) the load on

the engine is set by the user and the engine control system controls fuel injection

duration and timing to regulate piston motion and attempt to maintain steady

compression ratio. Single ended or opposed piston machines may also control gas

pressure in a bounce chamber to regulate piston motion. The problem with this

method is that there is a time delay of about one stroke before control action can be

taken. It also depends on low combustion variability. Mikalsen and Roskilly


consider piston control to be the main un-resolved challenge for the free-piston

engine concept and propose a predictive motion control method to reduce this time

delay and improve the transient response of the engine [88].

The second control philosophy is illustrated in Figure 1-2(b). The fuel quantity is set

by the user and the engine control system controls the load imposed on the mover to

regulate piston motion and attempt to maintain steady compression ratio. This

method results in demonstrably superior piston motion control since irregularities in

motion can be compensated almost immediately by a change in load, and the control

is not reliant on predictable combustion. However it requires the generator or pump

to be a controlled machine, not a passive machine.

Figure 1-2 Two control methodologies

Free-piston

engine

User set load

Fuel etc Piston motion

Control

Free-piston

engine

User set fuel

load Piston motion

Control

(a)

(b)


1.4 Discussion on free-piston engines state of the art

Despite the variety of free-piston concepts, all of the practitioners of this field are

attracted to the potential benefits of free-piston engines which can be summarised as:

Intrinsic variable compression ratio

Ostensible mechanical simplicity

The intrinsic variable compression ratio is due to the nature of the free-piston which

is un-constrained. In comparison to the kinematic nature of a crank driven piston,

whose path is determined by the geometry of the mechanism, the free-piston follows

a kinetic trajectory which can be adjusted by both changes in cylinder pressure and

load. Compression ratio is a critical parameter in the combustion process, and the

opportunity to choose an appropriate compression ratio for differing fuels and load

points is very attractive. Robust control of compression ratio has already been

demonstrated by several machines (eg Pempek, U.S. EPA [32]) but control of

machines without load control still appears problematic. A commonly stated aim of

free-piston prototype development is to implement Homogenous Charge

Compression Ignition (HCCI) which offers high efficiency and low emissions. This

requires fairly precise control of cylinder conditions to control the combustion

timing, and variable compression ratio can be used to accomplish this (providing that

accurate compression ratio control is achievable). Not only can combustion be

easily adapted across the load range using variable compression ratio, but variations

in fuel quality and even fuel type can be accommodated too. The fact that free-

pistons spend significantly less time near “top dead centre” than conventional

cranked pistons means that higher compression ratios are possible before the onset of

auto ignition, and heat transfer and NOx production are likely to be lowered [87]

(though combustion must be fast enough to avoid “time loss”). Finally, several

researchers are advocating very high compression ratio combustion (CR>30:1) as a

path to efficiency improvement [108, 109, 121].

The mechanical simplicity of free-piston engines has been noted by many

researchers, though it must be said that this does not necessarily translate to simple

construction. Although there are only one or two main moving parts, many engines

also have various valving requirements. For electric free-piston engines, a low

weight, efficient linear electric machine remains a technological challenge. Support,

lubrication and cooling of the piston assembly has received little if any attention in

the literature, but as more projects get beyond the concept stage and begin running


prototypes for long periods, lubrication and wear will become an important issue for

the mechanical design. Lubrication of piston rings is problematic for ported engines

(which most reported engines are), and may lead to increased burning of lubricating

oil in the cylinder. On the other hand, free-pistons have relatively low side loads to

support, and as several researchers have pointed out, higher compression ratios can

be more easily achieved with free-pistons compared to a cranked piston which

requires large bearings and stiff blocks and shafts to constrain the piston at high

compression ratios. Another possibility that has only been superficially explored in

relation to micro engines is un-lubricated ringless pistons that rely on very close

piston-cylinder tolerances and alignment for gas sealing.


1.5 Free-piston Engine Modelling

There is little that makes modelling of free-piston engines distinct from modelling

cranked piston engines. The only significant difference is due to piston motion.

Prescribed piston motion based on crank geometry is the common method for

cranked engines, and this is sufficiently accurate for most purposes. However free-

piston motion is a complex function of cylinder pressure(s) and instantaneous mover

load, so a truly predictive model needs to solve the piston dynamics coupled with

cylinder pressure and load.

Most modelling tools developed for cranked engines can be successfully applied to

free-piston engines, though certain empirical equations may need to be modified.

Nevertheless, the following section will summarise some notable efforts at

modelling free-piston engines in recent times.

1.5.1 Piston Dynamics Model

The motion of the rigid piston or mover assembly can be described using the

momentum equation as

where is any force that is exerted on the mover. Integrating WRT time yields

change in velocity, and integrating again yields change in position. Since the various

forces that act on the mover are complex and depend on non-linear influences such

as such combustion, valve/port opening and load behaviour, it is inadvisable to seek

an analytical solution. Rather, the problem can be conveniently solved on a

computer on a time-stepping basis where each force is evaluated instant by instant.

Since the initial value of velocity influences the solution, the solution will in general

vary somewhat for each cycle. If the dynamic behaviour of the engine model - run at

steady settings - tends toward stability, then the mover trajectory will settle to a

steady state after a number of engine cycles. Shoukry [105] reports his engine model

converged to a steady state after approximately 50 cycles, however my own

experience with the Pempek free-piston engine model was for around 3-6 cycles.

After the cylinder pressure, the most influential force on the mover is the load force.

Goldsborough and Van Blarigan [59] model the linear alternator as a force

proportional to velocity where the proportionality constant is constant throughout a

cycle but can be changed for different operating conditions. The WVU free-piston


engine model introduced by Petreanu [97] modelled the alternator load as a

sinusoidal function of position with a variety of shapes. The function was designed

to extract a set amount of work per stroke, which could be set by the user.

Friction has normally been found to be a small contributor to the mover motion,

owing to the low side force on the pistons compared to conventional engines.

Goldsborough and Van Blarigan [59] model friction as a combination of static and

viscous components with coefficients correlated to previous experiments. Shoukry

[105] describes a detailed friction model based on analysis of piston rings.

The monolithic liquid piston design recently developed at Vanderbilt University

[129] was modelled as a mass-spring-damper system, while the subsequent increased

inertance design used a flow based equation [123].

The largest contributor to piston motion is usually the cylinder pressure, especially at

the extremity of stroke when the pressure is very high. The determination of this

pressure is crucial for a viable engine model.

1.5.2 Simplified Cylinder Models

Single zone thermodynamic models form the basis of most free-piston engine piston

dynamics models. These treat the cylinder as a homogenous control volume which

is analysed using some variation of the energy equation. This is a powerful and

simple analysis which can yield accurate results. The accuracy of the model depends

on accurate specification of boundary flows, heat transfer, combustion rate and gas

mixture properties. A selection of single zone models is described below.

Goldsborough and Van Blarigan [59] modelled a free-piston engine using a

homogenous single zone. Combustion was modelled using a detailed chemical

kinetic mechanism in the hope of achieving better combustion model accuracy than a

reduced mechanism. The energy equation was written in terms of the species mass

fractions, internal energies and their rates of change. Combustion chamber leakage

was neglected. Heat transfer was by the Woschni correlation. Gas exchange was

modelled using equations for orifice flow, and a zonal formulation was employed to

model scavenging.

Petreanu [97] describes a cylinder model which operates in four differing phases.

Compression (until combustion begins) and expansion (after combustion ends) are

modelled as polytropic processes with a pre-determined polytropic index. The

combustion phase is modelled using the energy equation and assuming constant ratio


of specific heats. Heat transfer (presumably only applied during combustion) is

modelled using the Woschni correlation. Chemical heat release is modelled as part

of the heat transfer term and is prescribed with a Wibe function for mass fraction

burned. The fourth phase, the gas exchange period beginning with exhaust port

opening, was set to intake pressure, and the gas was instantaneously replaced with

cool, inlet air or air-fuel mixture (assuming perfect scavenging).

It is not immediately clear why the author used a polytropic exponent model instead

of the energy equation for the whole cycle, though perhaps it was deemed easier to

adjust to match experimental data. Regardless of the rational, this method restricts

the ability to use the model predictively, since the polytropic exponent represents an

empiricism requiring determination a priori. Also, the model ignores the gas

exchange process which, though bearing little immediate influence on piston motion,

effects important cylinder charge properties - namely residual exhaust fraction,

initial temperature and initial pressure. Nevertheless, this idealised gas exchange

model is sufficient to find an approximation of piston dynamics.

Larmi et al [74] report a cylinder model for a hydraulic engine using the energy

equation and accounted for inlet and exhaust flows using the nozzle flow equation,

heat transfer using the Woschni correlation and combustion using a triple-Wiebe

function. No mention was made of any gas blowby (leakage) model. The model

was capable of predicting the combustion cylinder and scavenging case pressures

with good accuracy, however air intake pressure had to be fudged higher, and

exhaust port pressure lower, since gas dynamic effects were apparently influencing

these processes. The model assumed fixed heat release and fixed hydraulic load, and

was rather less stable in operation than the real engine – with slight changes to the

load causing a cascading increase or decrease in piston velocity and compression

ratio. The author concludes that a more detailed hydraulic load model is needed.

Aichlmayr [12] developed a more sophisticated chemical kinetics single zone

cylinder model for modelling HCCI combustion in a micro free-piston cylinder. The

model used the energy equation and included the effect of blowby (which was

significant in this case). A good match with experimental data was achieved.

Mikalsen and Roskilly implement a simple cylinder model using the energy

equation, assuming constant specific heats and no blowby [80].

Single zone models have certain limitations because they cannot represent charge

inhomogeneities which affect combustion and emissions etc. Multi-zone models are


not common in free-piston modelling, perhaps because not many free-piston engines

use spark ignition; instead they typically employ either HCCI or direct injection

compression ignition. For greater combustion model fidelity in these cases a CFD

cylinder model is probably more appropriate.

1.5.3 Detailed cylinder models

The details of actual in-cylinder flows can be modelled using CFD, allowing

improved estimation of combustion chemistry and rate, emissions, heat transfer and

scavenging. However they are much more computationally intensive compared to

single and multi-zone models; and mesh setup can be complex and labour intensive.

Gas exchange modelling is also limited by the accuracy with which intake and

exhaust duct flows are modelled.

In an effort to find an optimum layout for scavenging, Goldsborough and Van

Blarigan [58] used the CFD code KIVA to analyse several configurations – loop,

hybrid-loop and uni-flow scavenging (Figure 1-3). Reasons for wanting good

scavenging were to prevent early ignition (due to added heat of charge mixture),

minimise fuel short circuiting and to ensure fast, reliable combustion.

Loop and hybrid-loop scavenging were both found to be unsatisfactory due to poor

scavenging performance, so a uni-flow layout was adopted. Optimum scavenging

was found with slow inlet velocity, zero port inclination and a swirl angle of 15

degrees. In the model, piston dynamics were prescribed based on previous 0D

modelling.

Kleemann et al [70] conducted 2D axisymetric and 3D simulations on the free-piston

engine of the FPEC consortium. They used a CFD code derived from KIVA-II but

considerably extended for modelling fuel injection and HCCI combustion. The

simulations were carried out iteratively between a 0D engine model, a 1D gas

dynamics model and the detailed CFD models.


Figure 1-3 Scavenging modes analysed by Goldsborough [58]

In a more detailed CFD analysis on the same project, Fredriksson et al [51] used

KIVA-3V to simulate the in-cylinder scavenging flows and combustion of a direct

injection diesel free-piston engine. A diesel oil surrogate model was implemented

into the KIVA code. Detailed chemical oxidation mechanisms of the two surrogate

fuels were modelled with 70 species in 306 reactions. Piston trajectory was

prescribed on the basis of previous 0D modelling. One significant finding was that

initial cylinder flow conditions (scavenging) had a large impact on subsequent

combustion, necessitating at least two or three complete cycles to gain realistic initial

conditions. Change of injection schedule had a significant influence on combustion

performance.

Mikalsen and Roskilly used the open source CFD code OpenFOAM to model the

combustion chamber of spark ignition free-piston engine compared to a conventional

cranked engine to analyse the difference in NO and CO emissions [81]. Initial

mixture after exhaust port closing seems to have been assumed homogenous and a

certain swirl velocity was prescribed. The piston trajectories were prescribed based

on previous analysis. Combustion was modelled using a flame area approach. The

concentration of NO and CO was modelled according to chemical kinetics equations

and for all other species, chemical equilibrium was assumed. A similar analysis was

also undertaken for DI diesel combustion [83]. These investigations included heat

transfer but no blowby. The model was subsequently extended to allow free-piston

dynamics to be directly coupled with cylinder pressure, allowing the effect of

combustion timing on compression ratio to be analysed [84]. The effect of an


insulated combustion chamber on emissions and efficiency is also investigated using

this model [87].

Mau et al [76] used commercial CFD code AVL_FIRE to model the cylinder and

“crank-case” of an electric free-piston engine under development to analyse the

differences in scavenging performance for a range of piston strokes and frequencies,

port positions and supercharging pressures. Piston motion was prescribed based on a

0D cylinder model and linear alternator electromagnetic model.

CFD models of the engine cylinder are dependent on the initial or boundary

conditions – in particular initial mixture distribution, temperature, velocity and initial

pressure. These can be predicted if the ducting leading up to the cylinder is

modelled, but the computational cost of extending the flow domain of CFD models

to the entire inlet and exhaust ductwork is prohibitive. A useful solution is to utilise

a simplified 1D gas dynamics code to provide time varying boundary conditions near

the cylinder inlets and outlets. 0D cylinder models are equally dependant on

boundary conditions, and here too a 1D gas dynamics model can be used to predict

cylinder inflows and outflows with greater realism than fixed pressure boundaries.

1.5.4 Gas dynamics

Reports of gas dynamics modelling in free-piston engines are few. This is perhaps a

reflection of the early state of development of many of these projects, where

predictions of combustion, scavenging, piston motion and valve timing have

assumed primary importance. However gas dynamic effects (especially in the two

stroke engine) tend to be highly significant, modifying flow rates through ports and

valves and affecting the final cylinder pressure. The breathing ability of a two stroke

engine can be dramatically improved and supercharging requirements reduced by

careful design of engine ducting. Goldsborough and Van Blarigan recognised the

importance of a gas dynamic analysis and planned to do it following a CFD

scavenging analysis [58]. A few 1D engine modelling efforts are outlined below.

Larmi et al [74] identified gas dynamic effects in experimental results of hydraulic

free-piston engine, compared to a 0D model of the same engine. A 1D gas dynamics

model was built using the commercial package GT-Power. Since this package is

designed for use with cranked engines, a number of work-arounds had to be

employed to enable it to model a free-piston engine. The piston motion had to be

specified in a lookup table as a function of crank angle degrees (CAD). This meant


that the piston motion was not dynamically calculated but constrained to a pre-set

trajectory, based on previous 0D simulations. The port openings likewise had to be

modelled as throttle valves opening as functions of CAD since the standard port

opening calculation was not suited to the asymmetric piston trajectory of a free-

piston engine. The simulated results of their work approximately followed

measured exhaust pipe pressure (Figure 1-4), thought it was unable to capture the

high frequency pulsations present in the measured data.

Figure 1-4 Simulated 1D model compared to experiment for pressure in exhaust

pipe. Larmi et al [74]

Fredriksson and Denbratt [52] also modelled the FPEC using commercial 1D gas

dynamics code BOOST. The cylinder model in the code was replaced by detailed

chemistry calculations from the SENKIN code.

Kleemann et al [70] mention a 1D gas dynamics model in their iterative 0D-1D-3D

modelling process, but do not give any details.

1.5.5 Other modelling approaches

Tóth-Nagy used a simulation model in conjunction with a combined genetic

algorithm-artificial neural network predictor model to explore the parameter space of

a free-piston engine for optimum combinations [115].

18

. . . .

19

Chapter 2 Pempek free-piston engine – details and

experimental results

2.1 Overview of the project

The Pempek free-piston engine is a symmetrical opposed cylinder, two stroke,

electric machine. It features an integrated compressor, overlapping generator

structure, and passive intake valves located in the head of the piston. [33, 96] The

design is targeted at compactness as it was envisaged for installation in hybrid

electric vehicles.

Pempek Systems Pty. Ltd. was engaged in developing this concept from about 2001

to 2008. During this period, the design of the original concept underwent minor

modification, and several versions were built and tested.

The first complete prototype used a single passive inlet valve in each piston, a single

electro pneumatic exhaust valve in each head, direct cylinder injection of gasoline

and spark ignition. The generator was a buried permanent magnet type, with a total

of eight poles on the mover and eight coils in the stator, each of which was actively

controlled by IGBT power transistors. A photo of the assembled spark ignition

prototype is shown in Figure 2-1. Engine testing commenced in early 2005.

Inconsistent combustion suggested scavenging issues, and the single large piston

mounted inlet valve was suspected of creating a large pocket of residual exhaust gas

in the cylinder. A four inlet valve piston was built to replace the single valve piston,

and the internal air space in the piston was reduce to ensure high scavenge air

pressure. At around the same time, a new cylinder head was introduced with a

central high pressure diesel injector and four exhaust valves.

This configuration proved to be much more successful at producing consistent

combustion. However, despite strong combustion and good indicated work, the

generator output remained stubbornly low. This was attributed to various loss

mechanisms in the generator, and a completely new generator was designed in an

effort to radically improve efficiency. The new generator employed a laminated

steel stator (the original used a powdered iron matrix), three times the number of

coils and a more traditional surface mounted magnet topology with back iron. The

same generator diameter was retained, however a decrease in the required magnet

Chapter 2 Pempek free-piston engine – details and experimental results 20

thickness allowed the combustion cylinder diameter to be slightly increased, and the

total mover mass was also decreased.

Figure 2-1 Spark ignition prototype

Unfortunately several problems plagued the new generator such as regular coil

failure, and lack of surface integrity of the inner stator surface, which caused rapid

wear of the mover bearings and imprecise positioning of the piston in the

combustion cylinder. Due to these problems, the final prototype was never

completely tested. Therefore, the following summary of experimental results will

draw on the successful engine runs from the original diesel configuration.

Two other un-tested modifications were proposed by the designer for this engine: A

low pressure passive inlet valve mechanism (see Figure 2-12), and an advanced low

energy electromagnetic exhaust valve actuator.

spark plug

generator stator body

gasoline injector

air intakes

exhaust valve actuator

cylinder head

generator coil terminals


2.2 Details of the engine

A cross section view of the Pempek engine is shown in Figure 2-2. Note that the

engine operates in a horizontal position, and is shown vertically here because of

space constraints. This diagram shows the new generator, however the

specifications listed below belong to the original diesel configuration.

2 cylinders per module

dimensions ~150x150x700 mm per module

Cylinder diameter 66mm

maximum stroke 112mm

typical stroke 104mm

typical geometric compression ratio 26:1

mover mass 5.97kg

operating frequency ~31Hz (1920rpm equivalent)

indicated power output ~12kW per module

instrumentation

o cylinder pressure sensor1

o compressor pressure sensor

o exhaust valve actuator cavities pressure sensor

o mover position2

o control signals log

The diesel version also had the following specifications

Common rail injector from Mercedes E320 CDI

common rail fuel pressure 200-1400 bar adjustable

1 The cylinder pressure sensor was an un cooled optic fibre type by Optrand mounted in the cylinder head. Published accuracy for combustion applications is 2% (FSO). The sensor package includes the signal conditioning unit, which outputs an analogue voltage or current proportional to measured pressure. Measured pressure data showed significant long term drift from one cycle to the next (cumulatively several bar), though this stabilised after the first few engine cycles. As a result, it was necessary to arbitrarily peg pressure each cycle to some assumed value. Roth et al [103] Roth, K., Sobiesiak, A., Robertson, L. A., and Yates, S., 2002, In-Cylinder Pressure Measurements With Optical Fiber and Piezoelectric Pressure Transducers, SAE, Paper 2002-01-0745 reported a detailed study of the Optrand sensor-signal conditioner package in combustion applications. They reported that it has a rather complex response to thermal shock, over-estimating the peak pressure, then under-estimating the remainder of the expansion stroke pressure. Peak under-estimation during scavenging could be as much as 0.5bar according to their results. 2 The calculation of move position was implemented by Pempek as part of the engine control system. The estimated accuracy was +-0.5mm.


Figure 2-2 Cross section of engine module (shown vertically)

exhaust valves

mover assembly

stator coil

cylinder sleeve

passive inlet valves

combustion cylinder

exhaust valve actuator

diesel injector

compressor intake check valve

compressor volume

diesel injector

exhaust valve armature

exhaust valve actuator coil

compressor cross-over duct

return springs

piston

cylinder head

cylinder head and exhaust port

air inlet

cylinder pressure sensor location

compressor pressure sensor location


The unbalanced vibration of a single two cylinder module can be cancelled out if

four modules are run together synchronously with opposite piston phasing.

The most unusual feature of the Pempek engine is the inlet air path. As shown in

Figure 2-2, air is drawn into the compressor through a non-return valve. The mover

assembly acts as a compressor and pressurises the inlet air. The compressed air is

delivered to a cavity within the opposite piston, before being released into the

cylinder via passive inlet valves in the head of the piston. This novel design feature

allows the engine to avoid using intake ports in the cylinder wall which can cause

accelerated piston ring wear and elevated oil consumption.

A significant space saving innovation is the overlapping of the large generator

magnet holder with the combustion cylinders.

A lot of effort went into the development of fast-acting electro-pneumatic exhaust

valve actuators. These were capable of opening and closing in 7-10ms. There were

some reliability problems however, relating to leakage of the gas spring cavities.

Furthermore, electric power consumption was quite high. (But new designs were

proposed to overcome these problems).

A typical opening trajectory of the quad exhaust valve armature is shown in Figure

2-3. The motion of the armature is inferred1 from the measured gas spring pressure.

Figure 2-3 Typical exhaust valve actuator trajectory

The exact motion of the exhaust valves themselves is slightly different to the above

trajectory, because they are spring mounted to the armature. (see Figure 5-22 below)

1 The closing gas spring chamber pressure, along with the known closed valve position volume of the chamber, allowed the valve armature position to be calculated by assuming polytropic compression/expansion and a polytropic exponent n=1.4.

-1

0

1

2

3

4

5

6

0 0.002 0.004 0.006 0.008 0.01 0.012

Lift

(mm

)

time (s)


In contrast to the exhaust valves, the intake valves (mounted in the piston head) open

passively when cylinder pressure is lower than compressor pressure. No direct

measurements were made of the opening trajectory of the intake valves, due to their

inaccessible location. However subsequent modelling shows their motion is highly

dependent on the combination of gas pressure forces in the cylinder and compressor,

and on the acceleration of the piston.

The exhaust valve opening and closing signals, fuel injection and spark timing (in

the case of the spark ignition configuration) were managed by a customised control

system, and were based on mover position. The position of the mover was not

directly measured but could be estimated based on generator coil voltage data.

The generator was an actively controlled machine, enabling it to regulate the motion

of the mover and control compression ratio. A typical velocity vs. position plot of

the mover is shown in Figure 2-4. The generator control system continually

monitored the mover velocity and position and adjusted the load force according to

whether the mover velocity was above or below the target velocity as

(2-1)

where is the instantaneous mover velocity and is a proportional control constant,

typically chosen to be about . The target velocity is determined as a

function of mover position as

(2-2)

where is the peak target velocity, is the target excursion from the

centred position, and the exponents and can be chosen to modify the shape of the

target velocity trajectory. The target trajectory is also shown in Figure 2-4. Note

that the centre position for the mover is nominated as the zero position.


Figure 2-4 Real vs. target mover velocity

This control method has proven to be very effective at controlling compression ratio.

The same control algorithm can be used for starting and motoring the engine.

A typical engine run is shown in Figure 2-5. The engine was first started and

motored for eight strokes (four cycles). Fuel injection began on the ninth stroke, and

the test ran for a further 13 cycles. Most of the engine tests were kept deliberately

short to protect the generator which (as mentioned above) was known to be

dissipating a large amount of energy.

-10

-8

-6

-4

-2

0

2

4

6

8

10

-60 -40 -20 0 20 40 60

velo

city

(m/s

)

mover position (mm)

target

measured


Figure 2-5 Example engine run indicated work (efficiency indicated by black

markers)

The indicated work for each cycle is shown in this example. While the engine is

motoring, heat and mass loss in the cylinder absorb around 50J per cycle. Once

firing, the energy of combustion produces positive work of around 150J per cycle.

Unfortunately, in this case, the left cylinder has a malfunctioning exhaust valve

which is effecting scavenging in this cylinder. On strokes 28 and 30, the valve

actually fails to open at all, and no combustion can then follow since there is no fresh

air in the cylinder. The important thing to note here is that despite wide fluctuations

in combustion energy in the left cylinder, the generator control system is easily able

to maintain correct compression. More detailed analysis of the engine performance

is presented below.

stroke

Indi

cate

d W

ork

(J)

Indi

cate

d ef

ficie

ncy

assu

min

g 14

.5m

g of

fuel

(%)


2.3 Cylinder pressure analysis

Figure 2-6 and Figure 2-7 show sample indicator plots from the right hand cylinder.

(See footnote on page 21 for details of the pressure sensor used, and Figure 2-2 for

sensor location) In this instance, injection of diesel fuel was at 300 bar (relatively

low) and beginning when the mover was 48mm from centre (about 1.6ms BTDC).

The fired cycle indicated work is 167 joules, or IMEP 4.7 bar, equivalent to about 10

kW for both cylinders. Based on the estimated fuel quantity of 14.5mg and a fuel

LHV of 43MJ/kg, the indicated efficiency is 27%.

The motored indicated work in this case was -57joules, or IMEP 1.6 bar, equivalent

to about -3.5 kW for both cylinders. Evidence here points to high leakage from the

combustion chamber, causing the high energy consumption for motoring, and

contributing to the poor indicated efficiency for the fired cycle.

Figure 2-6 Sample indicator plot – fired cycle

Figure 2-7 Sample indicator plot - motored cycle

0

10

20

30

40

50

60

70

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Pres

sure

(bar

)

mover position (mm)

167 J

0

10

20

30

40

50

60

70

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Pres

sure

(bar

)

mover position (mm)

-57 J


Figure 2-8 shows the net apparent heat release rate (HRR) from the cycle shown in

Figure 2-6. It was calculated very simply as

(2-3)

where the ratio of specific heats was assumed .

Figure 2-8 Apparent heat release

The above results can only be considered approximate, since a more rigorous

analysis would require modelling of the cylinder gas composition and temperature to

determined varying specific heats. Furthermore, actual combustion heat release

requires the effect of heat transfer and mass loss to be included. See Appendix III

for a further discussion on thermodynamic cylinder modelling.

The large dip in HRR just before the mixture ignites is mainly due to mass leakage

(though heat transfer and fuel evaporation also contribute). The main combustion

event lasts for approximately 3ms in total (relatively slow since the injection

pressure is only 300bar). Oddly, the results show continued heat release throughout

the expansion stroke, suggesting perhaps ongoing mixing controlled combustion (or

sensor drift may instead be to blame).

The estimated injected fuel quantity in this case was 14.5mg. Assuming complete

combustion, the actual heat release should be around 610J. Given that the apparent

heat release is only about 360J, it is clear that there are major losses - the most likely

-60

-40

-20

0

20

40

60

-100

0

100

200

300

400

500

-0.003 -0.001 0.001 0.003 0.005 0.007 0.009 0.011

posi

tion

(mm

)

HRR,

ene

rgy

(kJ/

s, J)

time (s)

mover position TDC

net apparent heat release (J)

heat release rate (kJ/s)


being incomplete combustion (due to insufficient oxygen and poor

mixing/evaporation), and mass leakage (blowby).

Figure 2-9 shows the cylinder pressure during the scavenging period. The exhaust

valves are seen to open rather early when the expansion stroke is about 70%

complete. This is followed soon after by the opening of the passive inlet valves, as

cylinder pressure falls below compressor pressure. Since the pressure sensor for the

compressor is located some distance from the intake valves (see Figure 2-2), there is

a short delay before the measured compressor pressure begins to fall. As fresh air

floods into the cylinder through the opening intake valves (from 13ms-16ms) there is

a momentary rise in cylinder pressure, followed by a dip. Due to the type of

pressure sensors used, the absolute pressure here is unknown and must be guessed.

Figure 2-9 Cylinder and compressor pressure during scavenging

Things to notice here

compressor pressure over three times atmospheric pressure

fluctuating cylinder pressure during scavenging (up to 0.4 bar)

unknown inlet valve trajectory (though it can be inferred to some extent)

The high compressor pressure was deliberately designed to ensure that the inlet

valves would open correctly and that scavenging would be strongly driven. A closer

analysis of the compressor is presented in the next section.

See also section 7.7.3 for results of modelling which show the probable inlet valve

trajectory and scavenging mass flows.

-60

-40

-20

0

20

40

60

0

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025 0.03po

sitio

n (m

m)

Pres

sure

(bar

)

time (s)

mover position

cylinder pressure exhaust valves open

compressor pressure

inlet valves open exhaust valves

close


Figure 2-10 shows a comparison of measured cylinder pressure during the

scavenging period with and without the exhaust pipe attached. All engine settings

were otherwise the same. Apart from a slightly higher initial cylinder blowdown

pressure in the no-pipe case (which is probably just coincidence) the only significant

difference without the pipe is that the cylinder pressure is almost steady. This

comparison illustrates the effects of gas dynamics in an exhaust pipe, even though in

this case the net effect on cylinder charging seemed negligible.

Figure 2-10 Comparison of cylinder pressure with and without exhaust pipe

0

1

2

3

4

0.01 0.015 0.02 0.025

Pres

sure

(bar

)

time (s)

With exhaust pipe

0

1

2

3

4

0.01 0.015 0.02 0.025

Pres

sure

(bar

)

time (s)

Without exhaust pipe


2.4 Compressor analysis

The integrated compressor and passive inlet valves are unique features of the

Pempek engine. Compressor characteristics (swept volume, clearance volume) can

be chosen by design, and though a high compression ratio ensured powerful

scavenging flows, the energy consumption was correspondingly increased. An

indicator plot of the compressor is shown in Figure 2-11.

Figure 2-11 Indicator plot of compressor

The work required for one compressor cycle here is about 48 joules, equivalent to a

cylinder IMEP of 1.35 bar, or about 3kW for both compressors. This is an enormous

loss of energy (almost 30% of the indicated work in the combustion cylinder).

Furthermore, the pumping work is dissipated in the inlet air stream, so assuming

around 440 mg of air is supplied each cycle (see Figure 7-80 below), the supplied air

will be heated above ambient by perhaps 100 °C (not counting heat transfer).

Clearly a reduction in compressor work was necessary.

A modified passive inlet valve mechanism was proposed by Pempek. This design,

as sketched in Figure 2-12 used a counterweight to counteract the natural inertia of

the valves, forcing them to open even with no gas pressure across them. This would

allow the compressor to operate at much lower pressure, while still providing

adequate air delivery. The valves are also tilted to induce strong combustion

chamber swirl. This mechanism has not, however, been tested as yet.

0

0.5

1

1.5

2

2.5

3

3.5

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Pres

sure

(bar

)

position (mm)

-48J


Figure 2-12 Proposed advanced passive inlet valve design [96]

A further concern for the original passive inlet valve configuration is the relatively

high seating velocity. Modelling suggests the valves bounce several times each

cycle, with seating velocities up to 3.5m/s (see Figure 7-79 below and [55]).


2.5 Summary

The most outstanding success of the Pempek free-piston engine project has been to

demonstrate robust and accurate piston motion control. The control method depends

on a controllable generator and is based on targeting a certain mover velocity -

which if met, will produce a predictable kinetic energy for compression. The control

algorithm can be easily made more sophisticated to suit the efficiency requirements

of the electric machine, for example, to make the typical force profile relatively flat

throughout the stroke. At the same time, it is able to instantly respond to unexpected

changes in combustion energy and keep the mover stroking to its prescribed

compression. A further useful capability of this control method is that it could easily

adjust the running frequency of the engine, which would be necessary if several

modules were run together to achieve dynamic balance. To re-iterate a point made

above in section 1.3, free-piston researchers should consider the demonstrated

control advantage of using a controllable load (be it a hydraulic pump or electric

machine).

On other points, the Pempek engine has been less successful, though this is hardly

unusual for such an ambitious mechanical design project, as the scattered literature

on free-piston engine design testifies (see Appendix I). Amongst all the engine

components developed specifically for this project, the linear electric generator and

electro-pneumatic valves actuators continue to challenge researchers around the

world.

The work reported in this thesis focused in the thermodynamic aspects of the engine,

specifically issues that effected power and efficiency.

Power – Early engine tests revealed substantially lower combustion energy than had

been expected. Conversion of the inlet valve system to four valves instead of one,

and using compression ignition instead of spark ignition improved combustion

energy somewhat, but it was still below expectation. In the ideal case (pure air at

room temperature and pressure), the cylinder would hold 430mg of air which would

should burn about 25mg of diesel fuel. Analysis of the engine’s actual fuelling limit

found that 14mg of fuel was about the maximum that could be efficiently burned.

Early analysis identified several factors that would reduce trapped air mass:

trapped residual exhaust (probably about 8%)

inlet charge heating in compressor (about 100 K)

late exhaust valve closing


late inlet valve closing (this was hard to confirm since the inlet valve

trajectory was unknown)

low initial cylinder pressure (this was hard to confirm since the cylinder

pressure sensors could not measure absolute pressure)

Taken together, these effects could easily account for the apparently poor charging

efficiency of the engine. Initial work focussed on the scavenging efficiency by

attempting to model the detailed in-cylinder scavenging flows with 3D CFD (see

animation titled “points” in Appendix XV for example). However, it was soon

realised that a more comprehensive gas dynamics analysis would be required to

investigate both the passive inlet valve trajectory, and the final cylinder pressure at

valve closing.

Efficiency – The high parasitic load of the integrated compressor was clearly

unacceptable, but some level of inlet pressurisation was probably necessary to drive

the scavenging process. Here again, a gas dynamic model of the engine was deemed

necessary to explore the lower limits of inlet charge compression, and the effect of

exhaust pipe gas dynamics on the charging process.

2.5.1 Motivation for the work presented in this thesis

Thus, this author deemed as crucial a comprehensive gas dynamic engine model for

investigating design options for improving power and efficiency. The model that has

been developed enables simulation of a complete engine cycle in around 2 minutes

on a desktop computer. The designer can use it to investigate:

optimal valve timing

optimal valve sizing

optimal inlet and exhaust duct shape

delivery ratio

necessary inlet charge compression etc.

The model is also useful for investigating piston motion and control issues which can

help the designer to correctly specify the electric machine (or hydraulic pump in the

case of a hydraulic free-piston engine), predict operating speed, evaluate control

methods etc.

The work presented here does not address detailed combustion physics, as indeed

this is an enormous field and has been investigated to some extent already by other

free-piston engine researchers (see section 1.5.3). Even though the piston trajectory


of a free-piston differs compared to conventional cranked engines, there is no reason

in principle why the many recent developments in combustion technology cannot be

equally applied to free-piston combustion.

2.5.2 Synopsis of remaining chapters

The next three chapters describe the important features of the engine model. Chapter

3 describes the cylinder model, which is a single zone thermodynamic control

volume. The model includes a chemical equilibrium calculation for modelling

combustion composition. Chapter 4 describes the gas dynamic model, which is

based on an existing technique, but with several modifications. Chapter 5 describes

miscellaneous other parts of the engine model which were not covered in the

previous two chapters. Chapter 6 explains the integration of all of the sub models

into a complete engine model structure.

Chapter 7 compares the gas dynamics model to a number of experimental test cases,

including a superficial comparison to measured data from the Pempek engine.

Chapter 8 serves as an application example of gas dynamic modelling in free-piston

engines. Firstly, using the completed gas dynamics engine model, the possibility of

lowering the compressor pressure of the Pempek engine is assessed. Secondly, a

radical modification to the Pempek engine is proposed and the model is used to

explore this possibility.

The final chapter summarises and concludes.

36

. . . .

37

Chapter 3 Thermodynamic and gas property models

This chapter details three key models that are applied to the engine modelling

problem.

Single zone thermodynamic cylinder model

Gas property model

Chemical equilibrium model

Models of engine cylinders range from single homogenous control volumes, to

multi-zone models and even to detailed CFD models with detailed chemistry.

However, since combustion rate modelling and emissions modelling are not part of

this study, a simpler single zone model has been adopted.

The cylinder model is carefully implemented to account for the properties of reacting

gas mixtures. Fuel oxidation rate must be prescribed, and can be set using an

appropriate empirical function. The cylinder model also allows mass transfer of

arbitrary mixtures such as fuel injection and piston blowby.

The second model detailed in this chapter is a gas property model which accurately

determines the relevant thermodynamic properties of a gas mixture based on

temperature and composition. This level of detail is necessary since gas properties

bear a significant influence on the thermodynamic model result. Furthermore,

accurate gas properties are also necessary for gas dynamic modelling (see Chapter 4)

The third model described here is a chemical equilibrium model. This models the

progress of chemical reactions during the burning and high temperature phase of the

engine cycle. This approach to modelling chemical heat release is advantageous for

a number of reasons: it ensures that changes in composition are accurately

represented; it automatically evaluates the heat release due to combustion for

arbitrary fuels and mixtures; and it models some important subtleties of high

temperature mixtures such as dissociation.

Chapter 3 Thermodynamic and gas property models 38

3.1 Thermodynamic control volume model

3.1.1 Energy equation for a control volume

The Law of the conservation of energy in a control volume in rate form can be

written as

(3-1)

where is the total enthalpy flow rate of any mass flows into or out of the

volume and will be written as henceforth. The equation can be expanded to

where the rate of work is due to change in volume. For a single species, the energy

equation can be written as

where is the total species mass change rate, including flows and chemical

reactions. By definition, the specific heat of the (ideal gas) species at constant

volume is . For a control volume containing several species, the equation

then becomes

Defining frozen mixture specific heat as

and noting that

and re-arranging yields

(3-2)


where the total enthalpy of any given flow is

(3-3)

where is the mass flow rate of each species in a flow across the system

boundary

For an ideal gas

Equation (3-2) describes an open control volume containing any number of reacting

chemical species, which is undergoing a change in volume with heat transfer.

Within the enthalpy term , only mass crossing the volume boundary is counted.

However the term for each species in the problem includes species mass

change due to both flows and chemical reactions. If chemical reactions are involved,

the values of species internal energy must be based on an absolute enthalpy scale

so that the relative enthalpies of formation of the various species are correctly

represented. See section 3.2 for more information on evaluating internal energies.

See Appendix III for a further discussion on using the energy equation in engine

modelling and pressure data analysis.

Note that the frozen specific heat is defined for fixed composition. See Appendix

II for a short discussion on this issue.

3.1.2 Numerical solution of the energy equation for a control volume

The most general and convenient description of the energy equation is given by

equation (3-2) since this places no requirements on the control volume to be non-

reacting or even to contain only ideal gas. Figure 3-1 illustrates the numerical

solution of this differential equation in T. The temperature and its rate of change are

known at time t1. An initial guess at the temperature at t2 T2a is made, after which

equation (3-2) is employed to evaluate dT2a/dt. Assuming a parabolic curve in T

between t1 and t2, a refined estimation of temperature T2b is found by

(3-4)


Figure 3-1 Numerical solution of the energy equation

The process of successively refining the estimation of T2 is repeated several times.

Since it is done in the context of an engine simulation, all the other terms in equation

(3-2) are also iteratively updated as part of the process. The sequence of operations

is described in Table 3-1. The initial guess is based on extrapolated data from the

previous time step. A conservative initial guess for T based on the assumption of

zero rate of change at the current timestep was used to guard against overshoot.

Computationally intensive evaluations such as combustion rate modelling, chemical

reactions, gas property calculations, heat transfer and piston motion were kept to a

minimum and only iterated twice. The second evaluation of pressure was averaged

with the first to guard against oscillatory results in the case of small control volumes

with large pressure dependant mass flows. Piston motion was only evaluated twice,

since the time scale of piston motion is typically much longer than pressure changes

in the cylinder, with the result that piston motion is not likely to deviate much from

the initial estimates. In the case of a conventional cranked engine, piston motion

may be considered fixed by crank-shaft geometry and therefore prescribed at the

beginning of the cylinder calculation with no further iteration necessary. In Table 3-

1, actions that are single spaced do not have to be carried out in order, while actions

listed after a space are dependent on the previous calculation or calculations.

t2 t1

T2a

T1

T2b

Time

Tem

pera

ture


Table 3-1 Thermodynamic cylinder model calculation sequence

Action Variables Iteration Guess piston motion (1) Initial guess at (conservative extrapolation) Initial guess at mass and species change

Update Set to the same as prev step Set to same as prev step

Evaluate (1)

Evaluate (1)

Update (1)

Evaluate piston motion (2)

Evaluate any combustion (1)

Evaluate any chemical reactions (1) Evaluate any mass flows (blowby, fuel injection) (1)

Evaluate mixture properties (1) Evaluate any heat transfer (1)

Evaluate (2)

Evaluate (2)

Update (2)

Evaluate any combustion (2)

Evaluate any chemical reactions (2) Evaluate any mass flows (blowby, fuel injection) (2)

Evaluate mixture properties (2) Evaluate any heat transfer (2)

Evaluate (3)

Evaluate (3)

Update (3)

Evaluate piston motion (3)


3.1.3 Validation of the thermodynamic model

A simplified test case was modelled using the solution of equation (3-2) as described

in section 3.1.2. The test case was a closed volume undergoing a sinusoidal

compression-expansion cycle with a compression ratio of 20:1. The contents of the

cylinder was an ideal gas with a constant ratio of specific heats and an initial

pressure and temperature of 1bar and 300K. The modelled pressure was compared

to the analytical solution for isentropic compression of a calorically perfect gas.

Figure 3-2 shows the exact pressure and the relative model error over one cycle for

the case of 200, 100 and 50 time steps. More timesteps improves the accuracy of the

model, but even with a relatively coarse number of time steps, the maximum error is

0.5%.

Figure 3-2 Pressure history and error for various time steps

A more complex test case was modelled which included a combustion event. The

same compression-expansion cycle and initial conditions were used as the above

case, but the initial cylinder contents were 21 parts Oxygen, 79 parts Nitrogen, 1 part

water vapour and 0.038 parts Carbon Dioxide, and 5 parts iso-octane by mass. The

specific heats, internal energy and chemical reaction rates of the gas mixture were

calculated using the methods described below in sections 3.2 and 3.3. Figure 3-3

shows the prescribed fuel burn rate, with ignition suddenly beginning at TDC, and

falling away asymptotically as the fuel is consumed. The fuel burn rate is given in

mass of fuel per mass of cylinder contents per second, and the total time for the cycle

was 20ms. A discontinuous fuel burn profile was selected to provide a severe test of

the thermodynamic model.

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

P (b

ar)

compression - expansion cycle

-0.6%-0.5%-0.4%-0.3%-0.2%-0.1%0.0%0.1%0.2%

0 0.2 0.4 0.6 0.8 1compression - expansion cycle

error

20010050


Figure 3-4 shows the pressure and temperature produced by the model for the case of

400, 200 and 100 timesteps.

Figure 3-3 Prescribed fuel burn rate and cylinder volume

Figure 3-4 Pressure and temperature for combustion case

Since an analytical solution for combustion was not available, the simulation using

400 time steps was used as a base line and the other two simulation runs with fewer

0

0.25

0.5

0.75

1

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

vol

fuel

bur

n ra

te k

g/kg

/s


df_dt

V

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

1.E+07

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P (Pa)


P 400P 200P 100

0

500

1000

1500

2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

T (K)


T 400T 200T 100


time steps were compered. Figure 3-5 shows the relative error in pressure or

temperature for the case of 200 and 100 time steps. The error is larger than in the no

combustion case and this is probably due to the discontinuous nature of the

combustion even. Even in the case of only 100 time steps, the error is mostly less

than 1%.

Figure 3-5 Error for various time steps

It is worth noting that much larger errors may occur if mixture properties are

incorrectly modelled (see Figure 3-7 below). The gas property model described in

the next section was developed in order to ensure that gas property errors are small.

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

erro

r


200

100


3.2 Gas mixture property model

Very simple thermodynamic models sometimes use the specific heats of air at room

temperature. However the assumption of constant specific heats may lead to large

errors in engine simulations, since the specific heat of engine gases changes with

temperature, as can be seen in Figure 3-6. Moreover, the specific heat of any given

mixture of gasses will depend on the relative quantities of each gas in the mixture.

Figure 3-6 Specific heat Cp of common exhaust gas species

As a test case, a closed volume undergoing a sinusoidal compression-expansion

cycle with a compression ratio of 20:1 was modelled for both constant specific heats

( =1.4) and for the specific heats calculated with a typical atmospheric mixture of

gases (as described below in section 3.2.2). The constant specific heat case over-

predicted the peak pressure by about 7%, as shown in Figure 3-7.

Figure 3-7 Typical pressure error incurred for setting =1.4

0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000

0%

2%

4%

6%

8%

0 0.2 0.4 0.6 0.8 1

erro

r


H2O

N2

CO2

O2

T (K)

Cp (

J/kg

/K)


A simple thermodynamic model can be significantly improved by dividing the

cylinder process into periods of compression, combustion and expansion and

assigning to each a ratio of specific heats or polytropic exponent that is constant

or linear in temperature [62]. This method tries to account for the variation in gas

properties that occur throughout the engine cycle, and it can also be tuned to allow

the effects of heat transfer and gas leakage to be crudely but effectively modelled.

The best value for is determined based on experience, so this model is not well

suited to predictive simulations.

Krieger and Boorman [72] introduced a model of using mathematical expressions

which are functions of temperature, pressure and equivalence ratio and are valid for

any fuel with a composition of the form . Klein [71] explored various models

for in an engine cycle simulation against a full chemical equilibrium model and

found that while a linear function in temperature is sufficiently accurate for

unburned gas up to a moderate temperature, a polynomial expression such as by

Krieger and Boorman is necessary for burned gasses. Klein proposes a two zone

model consisting of unburned fuel-air mixture and burned gas. Mixtures involving

fuels with differing ratio (to the curves derived by Krieger and Boorman) will

need to be analysed for chemical equilibrium properties and have new sets of curves

fitted. To illustrate the typical variation in , the ratio of specific heats for air-fuel

mixture and for combustion products is shown in Figure 3-8.

Rather than conceiving of engine mixtures as either burned or unburned, better

generality can be achieved with a small increase in computational burden by

considering all of the significant gas species, and this is the approach taken here.

Using this approach, fuels of arbitrary composition can be specified, as can differing

air humidity, EGR, equivalence ratio etc. Also importantly, chemical reactions such

as dissociation, combustion and exhaust after-treatment may be modelled.


Figure 3-8 Variation of with temperature and equivalence ratio for unburned and burned mixture 1

3.2.1 Species properties

The gas properties of typical products of combustion are taken from the JANAF

thermochemical tables [38] which list the enthalpy and specific heat on a per-

mole basis. These are listed in Appendix IV in Table IV-1 and Table IV-2.

Use of tabulated data may be more computationally efficient than using polynomial

curve fits, since each temperature increment can be made to correspond to an array

index from 1-60. Values are then simply interpolated linearly between the two

nearest temperature values.

Properties of various common fuels are listed in Appendix IV in Table IV-3.

The gas species are presumed to be ideal gases, so the internal energy is found by

1 is the ratio of frozen specific heats – see Appendix II for a short discussion

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

g*

Unburned mixture by equivalence ratio

0

T (K) Mixture made from 75 parts N2, 15 parts O2 and various parts n-octane (C8H18) by mass

g*

Burned mixture by equivalence ratio

0.2 0.4 0.6 0.8 1

0 0.2

0.4 0.6

0.8 1

100 bar

1 bar


Where the universal gas constant (J/mol/K)

The specific heat at constant volume is found by

The energy properties can be converted from per-mole to per-kilogram quantities by

dividing by the species molar mass M (g/mol).

3.2.2 Mixture properties

Mixture properties are calculated as

where the mixture contains any number of chemical species s, and is the mass of

each species. The ratio of specific heats for the mixture is

The mixture’s gas constant is

The starred notation is used here to indicate that these are frozen mixture quantities

and are not in general equal to the true specific heats (when shifting chemical

equilibrium is allowed to cause changes to species fractions). See Appendix II for a

short discussion on this issue.


3.3 Reacting gas mixture model

At low temperatures, mixtures of gases typically do not react so any initially

specified composition will remain constant and the mixture properties can be

calculated directly as described above in section 3.2.2. However at elevated

temperature, the initial mixture composition may change due to chemical reactions.

Combustion reactions at low temperatures (T<1400) can be evaluated analytically,

however at higher temperatures significant extra species may be produced and must

be solved numerically. The importance of these extra species is twofold. Firstly, if

present in large enough fraction, they will affect the energy balance of the mixture,

altering the mixture enthalpy and specific heat. Secondly, even though some species

may not be in high enough fraction to significantly alter the energy properties of the

mixture they may nevertheless be important species to consider as pollutants.

After the onset of combustion, the temperature is usually high enough and reaction

rates fast enough that gas mixtures can be assumed to be in shifting chemical

equilibrium [62]. An exception is the oxides of nitrogen.

3.3.1 Chemical equilibrium model

The model used here uses the concept of equilibrium constants. This method for

calculating equilibrium composition is commonly used [46, 94, 99, 104, 106, 127]

and helpful explanations can be found in [30, 49]

Species considered in the equilibrium calculation are listed in Table 3-2. The

species in this list are the only species to exist in significant quantity at common air-

fuel ratios, temperatures and pressures found in engines. Indeed in normal

circumstances N could probably be neglected since it is rarely present above 1 part

per million. It is included here because it is involved in the reactions that produce

and consume NO. Note also that in very rich mixtures (carbon:oxygen atom ratios

approaching 1), a large number of carbon based species will form.


Table 3-2 List of species considered in equilibrium calculation

Name Chemical formula carbon monoxide CO carbon dioxide CO2 hydrogen, monatomic H hydrogen, diatomic H2 water vapour H2O nitrogen, monatomic N nitrogen, diatomic N2 nitric oxide NO oxygen, monatomic O oxygen, diatomic O2 hydroxyl OH

The mixture contains a set number of atoms. We can write four mass balance

equations for moles of atomic carbon, hydrogen, nitrogen and oxygen.

Where is the mole fraction of each species and is the total moles of the

mixture. Thus there are 12 unknowns - 11 species mole fractions and the unknown

total moles. Another mass balance equation can be written for the sum of all species

mole fractions, which by definition is 1.

More equations are needed to solve the 12 unknowns. Table 3-3 contains a list of 8

suitable equilibrium equations. The thermodynamic property (Gibbs free energy)

is defined as

where is the molar specific entropy at the reference pressure (0.1Mpa) and is

the absolute enthalpy. P is in bar.

Equilibrium equations 1,2,4,5,6,7,8 were used, along with the five mass balance

equations. Details of the numerical solution that was devised are given in Appendix

V.


Table 3-3 Equilibrium equations

Reaction Equilibrium equation Equilibrium constant

1

2

3

4

5

6

7

8

To speed up equilibrium computations, the equilibrium constant for each of the eight

reactions was pre-calculated for temperatures between 100 and 6000K. The values

are listed in Appendix IV in Table IV-5

3.3.2 Validation of the equilibrium model

The equilibrium model was compared to the industry standard NASA code CEA

[79] for a range of test cases. Figure 3-9 shows the equilibrium mass fractions

resulting from a set mixture for varying temperature and pressure. Figure 3-10

shows the equilibrium fractions at fixed temperature and pressure for varying fuel to

oxygen ratio. The mixture chosen for analysis is somewhat arbitrary; however it is

representative of the conditions found in an engine. The model corresponds well to

the NASA code results, even though the NASA code considers a comprehensive list

of species, while the model here is restricted to 11 specific species. Potentially

significant mass fractions of NO2 are apparent in the NASA results on the lean side

of Figure 3-10, and ammonia and several carbon based species begin to appear at

very rich mixtures. It is also apparent that at low mole fractions the NASA code

suffers from some kind of rounding error, but this is hardly significant.


Figure 3-9 Equilibrium species mass fractions of a fuel-air mixture at various temperatures and pressure

Figure 3-10 Equilibrium species mass fractions of a fuel air mixture with varying fuel to oxygen ratio

0.000000

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1500 2500 3500 4500 55000.000000

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1 4 16 64 256

0.000000

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0% 10% 20% 30% 40% 50% 60% 70% 80%

H2

H2O

N2

OH

NO

CO CO2

O H2O

N2

CO2

N

H

N

O2

CO2

CO

NO

OH O2 H2

O

P=1 bar T=3000K

Mixture made from 75 parts N2, 15 parts O2 and 5 parts n-octane (C8H18) by mass

3

dT (K)

C H ) byP (bar)

Mass fraction H

Present code

NASA code Present code

NASA code

H2O

N2

OH

NO

CO

CO2

O H

O2

H2

P=50 bar T=2000 K

NO

OH

CO

H2

H2O

CO2

n-octane (C8H18) to oxygen (O2 ) ratio (kg/kg)

Mixture made from 75 parts N2, 15 parts O2 and varying parts nOctane (C8H18) by mass

Mass fraction

O2

NO2

CH4

HCN

NH3

Present code

NASA code


3.3.3 NO Rate limit

Due to the importance of NO as an atmospheric pollutant, the model includes a rate

limiting adjustment for NO production/destruction according to the extended

Zeldovich mechanism as described by Borman [30] and Ferguson [49]. This is

necessary because the rates of chemical reactions producing/consuming NO are

significantly slower than the other reactions and the equilibrium assumption is not

accurate in its case.

3.3.4 Notable features of the equilibrium code

Robust for all ratios up to about 0.95. Past this point, the O2 fraction

becomes too small to accurately compute. The only carbon compounds

considered are and , so past this point, excess fuel is retained as un-

reacted fuel in the mixture

Zero hydrogen, carbon and nitrogen are permitted; atomic molar fractions

less than 5x10-6 are rounded down to zero for the sake of numerical

expediency. Only oxygen must be present at a finite fraction.

The code is efficient, reaching adequate accuracy with an average of five iterations

each involving a 4x4 matrix division.

54

. . . .

55

Chapter 4 Unsteady 1D gas dynamics model

4.1 Introduction

Gas dynamic effects on reciprocating IC engines have been recognised for many

years, however gas dynamic effects are not always considered in engine models. A

simple idealised example will suffice to demonstrate the potential modelling errors

from ignoring gas dynamics effects. The mass flow rate out of a reservoir through a

suddenly opened duct is shown in Figure 4-1. The duct is 300mm in length and

connects two reservoirs with a pressure ratio of 1.4. The duct is initially at the

downstream pressure, and is frictionless and adiabatic. This case is similar to

exhaust blowdown, when a valve or port opens rapidly and gas at high pressure

flows into the exhaust manifold.

Figure 4-1 Evolving mass flow rate into an idealised duct

The mass flow from the upstream reservoir takes several milliseconds to establish a

steady value due to the time it takes for pressure changes to propagate to the end of

the duct and back again several times - the longer the duct, the longer the flow

transient. If this flow were modelled as quasi steady (neglecting gas dynamic

effects), the mass flow rate would be greatly over-predicted for the first 2ms or so.

Admittedly, the effects of gas dynamics in real engines are not always as dramatic as

in the example given above. A low speed four stroke engine exhaust stroke may

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10ms

Gas properties R=278J/kg/K, =1.4, P1=1.4bar, T1=330.3K, P2=1bar, T2=300K

Duct is 300mm long Results from a gas dynamic model

Mas

s flo

w ra

te %

of s

tead

y flo

w

Chapter 4 Unsteady 1D gas dynamics model 56

take 30ms or more to complete (far longer than the ~2ms taken for the example

above). Often the flow area at the engine valves is somewhat smaller than the

exhaust pipe, and this further militates against gas dynamic influences.

Nevertheless, gas dynamics is known to have significant influence on four strokes

engines, especially high performance engines [95], and both inlet and exhaust

ducting contribute to the ability of the engine to ‘breath’.

Two stroke engines are even more influenced by gas dynamics effects because the

gas exchange period is shorter than a four stroke, large port areas are typical, and gas

exchange is driven by pressure differential across simultaneously open inlet and

exhaust ports, rather than the positive displacement pumping of the four stroke cycle.

A common gas dynamic strategy for increasing trapped mass in two strokes is to

design the exhaust pipe so that a plugging pulse arrives at the exhaust port just

before it closes. Design of exhaust and inlet ducting to maximise the benefits of

natural resonances is notoriously difficult when the engine must operate over a range

of speeds. A compromise is inevitable between high peak power on the one hand

(aggressive tuning for a narrow RPM band), and drivability on the other (a flat

torque curve).

Two stroke free-piston engines such as the Pempek engine are uniquely positioned to

take full advantage of gas dynamic tuning, because they are essentially constant

speed (i.e. frequency) machines. One of the central aims of this modelling project

has been to explore the potential of gas dynamic tuning for the Pempek engine.

4.1.1 Overview of gas dynamic modelling for IC engines

The relationship between pressure and velocity for a pressure wave travelling in one

direction was derived by Samuel Earnshaw [44] in 1860. Bannister and Mucklow

[18] experimentally validated the theory of finite waves using a shock tube

experiment. Riemann [101] had earlier proposed the method of characteristics

(MOC) and this was subsequently developed into a graphical solution for unsteady

flow problems. Benson et al. [22] used digital computing and the MOC to evaluate

unsteady flows in engine ducts. By the 1980’s gas dynamics in engine modelling

using the MOC was well established [21]. Since then finite difference formulations

have become popular for engine gas dynamics [124], though work using the MOC

continues eg [132].


Despite a long history and the feeling by some that it is time to close the book on

further development in this field [124], diverse, high quality work on unsteady 1D

gas dynamics continues to be reported in the literature. Moreover, there remains an

incredible variety of methods and codes in use in engine research laboratories around

the world that reflect the particular historical needs and emphases of each research

group, and the pedigree of its gas dynamics code, whether it be a MOC based

formulation, other wave action method [24], or a finite difference formulation.

Much of the recent work reported in literature relates to improving boundary or

junction models (eg [19, 29, 36, 54, 92, 111]). Accuracy in modelling boundaries is

critical in engine modelling, since there are numerous duct boundaries that trace the

path from air inlet through the combustion cylinder to exhaust outlet. Furthermore,

new opportunities in detailed CFD present new challenges in interfacing. Since

boundary formulations are typically based on wave action concepts (even finite

difference codes), work reported in this area has relevance for all 1D gas dynamics

codes.

4.1.2 Requirements for the gas dynamics code

A two stroke engine is highly influenced by gas dynamics, and due to the nature of

these engines (sharp flow events, cool air short circuiting, tapered ducts, ports,

valves, crank cases and the like) the code must be good at dealing with these kinds of

flows. A code capable of modelling difficult flows will also excel at modelling less

challenging engines. A code was developed to model the Pempek free-piston engine

with the following requirements in mind.

Mass conservation – not many codes are strictly mass conservative and even

though small discrepancies may be tolerated, any error here will probably

produce an error of the same proportion in engine or turbine/compressor power

predictions [69].

Tapered ducts – tuned two-stroke exhaust pipes usually employ long tapered

sections, so accuracy here is important.

Sudden area changes – There are numerous locations in within the engine’s flow

path where sudden changes in flow area occur. Ports and poppet valves are time

varying flow restrictions.

Gas property discontinuity – Changes in gas temperature and composition within

the passages of the engine have a strong bearing on the passage of pressure


waves. Regardless of whether the variation in properties is sharp or diffuse, the

effect on pressure waves must be properly accounted for.

Flow losses - Heat transfer in exhaust ducts can be significant, as can the

resulting change in gas temperature. Friction has a small effect in straight ducts,

but can also be used to model separated flow such as in bends and downstream of

restrictions.

Numerical damping – first order discretisation such as is typical of the mesh

MOC tends to smear details in the flow. Compared to a second order scheme and

depending on the wavelength of typical pulsations, more mesh points may be

needed to achieve satisfactory resolution. On the other hand, higher order

schemes must not produce spurious oscillations at sharp changes in gradient.

Computational efficiency – One important advantage of 1D gas dynamics models

over multi-dimensional CFD models is brisk execution. The faster the code, the

more efficiently the engine designer can work through the iterative process of

design and simulation.

Code complexity – this impacts the usability of the code; the ease by which it can

be de-bugged, errors discovered, functionality added or modified and the number

of people who can make use of it.

Moving and deforming ducts – not an essential capability, but moving piston

problems such as the combustion and compressor cylinders of the Pempek engine

are usefully modelled using deformable ducts.

4.1.3 Rational for writing an in-house code

There were a number of commercial engine simulation programs available. The

most prominent were:

Virtual 2/4 stroke – Optimum power [4]

Wave – Ricardo [5]

Boost – AVL [1]

GTPower - Gamma Technologies [2]

Lotus engine simulator - Lotus Engineering [3]

This author decided to write his own gas dynamics code for the following reasons.

Firstly, these packages are designed for conventional cranked engines, utilising cam

driven poppet valves or cylinder ports. It was uncertain how easily they could be

adapted to the unique requirements of the Pempek free piston engine, with its un-


constrained piston and inlet valves. The integrated compressor also presented

special difficulty, due to its dynamic change of length, and rapid acceleration.

Secondly, there was limited information on the actual simulation methods, and

assumptions used in the commercial programs. They are essentially “black boxes”,

since the average user is not interested in how the results are produced, and

furthermore, the code is proprietary information. Thirdly, commercial licencing was

found to be very expensive. It therefore seemed preferable to maintain control of

this aspect of the engine model.

4.1.4 Summary of the gas dynamics code

The 1D gas dynamics code this author developed is based on the method developed

at Queens University, Belfast eg. [23, 50] and subsequently adopted by the

commercial simulation package Virtual EnginesTM. That method has been re-worked

by this author. The original first order (linear) interpolation of pressure waves

extended to second order and the way heat transfer is incorporated is modified and

includes a method for achieving full mass conservation.

In the model, each computational cell is treated as an idealised constant-area,

constant-property, frictionless duct, so that simple algebraic expression are sufficient

to describe the passage of finite amplitude waves. Area change, gas property change

and friction are accounted for at the interfaces between cells. Connection at duct

ends is treated in exactly the same way. Thus the model uses a uniform theoretical

treatment of boundaries throughout. The boundary solution algorithm summarised

here was developed with careful consideration to the accuracy, numerical efficiency,

and stability of the solution for a wide range of possible flow cases.

This method is similar to the mesh method of characteristics. Common criticisms of

these older wave action methods compared with newer finite difference methods are:

numerical smearing due to first order discretisation, poor mass conservation, and

high computational load. The code presented here addresses all of these issues.

Numerical smearing is reduced by second order discretisation and a method is

introduced to effectively maintain mass conservation. Finally, computational load of

any engine gas dynamics model depends strongly on the number of complex flow

boundaries that need to be solved (which are many). The present method lays great

emphasis on accuracy and efficiency of these boundary flow solutions. The elegance

of finite difference solutions is less obvious when all of the necessary flow

boundaries and gas property variations in a real engine are considered too.


Validation of the model is described in section 4.6 and Chapter 7. Direct

comparison with other popular numerical schemes is not included here, however

useful comparisons can be found in work by others such as [37, 42, 69, 124].


Position

Time

Low pressure

Gas in a frictionless duct

Particle path lines

Pressure wave High pressure

4.2 Theoretical Basis

The foundational equation used here for evaluation of unsteady gas flow is that given

by Earnshaw [44] for a pressure wave travelling in one direction where a particle

experiences a change in velocity as a function of the change in pressure.1 The

equation assumes a calorically perfect gas.

(4-1)

Where and are the final and initial velocities, and are the final and initial

pressures, is the ratio of specific heats and is the speed of sound at the original

pressure . The compression or expansion process is assumed to be isentropic, so

the final speed of sound is found as:

(4-2)

In equation (4-1) the positive direction of velocity is in the same direction as the

motion of the pressure wave. The physical meaning of the equation is illustrated in

Figure 4-2. Particles in 1D flow are accelerated toward the right by the influence of

a right travelling pressure wave.

Figure 4-2 A right travelling pressure wave

1 A derivation of equation (4-1) is presented in Appendix VI


Figure 4-3 Oppositely moving pressure waves

Setting the quiescent velocity in equation (4-1) to zero and evaluating the general

case of oppositely moving pressure waves as shown in Figure 4-3, equations for

velocity and pressure as functions of the left and right travelling pressure waves can

be written as:

(4-3)

(4-4)

Velocity is positive in the rightward direction. The variable is shorthand for

(4-5)

The subscripts R and L signify the rightward and leftward travelling pressure waves

respectively. Where appears without a subscript, this signifies the superposition

pressure – i.e. the static pressure. Note the reference pressure can be set to an

arbitrary value, though it should be close to the typical pressure being modelled. It is

conventional to set it to atmospheric pressure.

P

Position

PR u PL u

u0 P0


4.3 Wave Propagation

Earnshaw’s equation (4-1) was derived under the conditions of constant flow area,

constant gas properties and no friction or heat transfer. If these conditions are met,

then the waves will propagate through the gas with unchanged magnitude (though

they may distort due to uneven propagation velocities). At any given instant, the

value of the left and right travelling pressure waves at all points along the duct can

be derived from the local fluid velocity and pressure by re-arranging equations (4-3)

and (4-4)

(4-6)

(4-7)

The left and right travelling waves are then advanced on a timestep basis where the

speed of propagation at each point on the wave is the sum of the local speed of sound

and the flow velocity.

(4-8)

Figure 4-4 illustrates the procedure for advancing both waves. The wave speeds

were calculated using the flow velocity and pressure at the diagonally opposite node

along with the mid cell fluid properties. A common timestep is used for all ducts

and volumes in a model. In the engine model, a value of 0.1ms was used.

Figure 4-4 Advancing pressure waves by one time step

The value of the wave incident on the mesh points at the current time step is found

by interpolation of the previous mesh point values. Details are given in the next

section.

time

current time-step

previous time-step

Left travelling wave

Right travelling wave

position


4.3.1 Second Order Wave Interpolation

In typical mesh MOC solutions this interpolation is linear (or first order) between the

two adjacent mesh points. This has two disadvantages. First, unless the wave

traverses exactly one mesh space in one time step, successive interpolations result in

smearing of features. Second, the distance traversed must not be allowed to exceed

one mesh space, since this would result in extrapolation and solution instability. As

a result the majority of waves in a model will tend to traverse much less than the

ideal one mesh space. The so-called Courant number is the proportion of a mesh

space traversed by a wave in one time step.

To avoid these problems, this author devised a second order wave interpolation

scheme. A third ‘upwind’ mesh point is included in the interpolation procedure, so

that a parabolic curve can be fitted between these points. This improves the

resolution of the model, and permits waves to traverse somewhat more than one

mesh space during a time step. This is illustrated in Figure 4-5.

Figure 4-5 Second order interpolation of pressure waves

Numerical overshoots will appear in the solution near large changes in gradient,

unless precautions are taken. The method adopted here is to simply limit the value

returned from the interpolation to between the upper and lower values in the interval.

If there is any friction, area change or the like at the middle node (which would

modify the value of the incoming pressure wave), then the value of the wave at the

far upwind node must be modified accordingly. Furthermore, if the far upwind node

is outside the duct, then an alternative estimate is required. Details are in Appendix

IX.

x0 x1 x2

Basic curve – three point quadratic

Non-overshoot value limit

position

wave


4.3.2 Heat Transfer and Mass Conservation

The basic theory of wave propagation outlined above does not allow for heat transfer

to or from the walls of the duct, or indeed for heat released internally due to

chemical reactions. The approach that was developed treats each section of duct

between mesh nodes as a control volume (computational cell). Midway through

each timestep, heat transfer is applied to the gas in each cell, which results in a small

instantaneous change in local pressure, but no change in local velocity. The situation

is illustrated in Figure 4-6. If the cell pressure must be adjusted higher, then both

right and left travelling waves are adjusted higher (by an equal amount), and vice

versa. The steps are as follows:

Figure 4-6 Modifying pressure waves to account for heat transfer and mass

conservation

Step 1 Calculate the mid cell, mid time step pressure before heat transfer is

accounted for.

Step 2 Calculate the mid cell pressure that occurs when heat transfer is accounted

for.

Step 3 Alter the left and right travelling waves according to the change in pressure.

There are two possible ways to estimate the initial mid cell pressure. The first way is

to link the mid cell pressure to the pressure wave values (which are known at the

mesh nodes), and estimate the left and right travelling wave at the cell centre. The

other way is to link the mid cell pressure to the cell mass via the ideal gas relation

. However in the standard implementation of wave action methods, cell

mass is not explicitly linked with local pressure. Thus either mass must be corrected

at each timestep to bring it into line with the local pressure, or the pressure waves

must be modified each timestep to bring the local pressure into line with mass

conservation. The former method is typical - hence those codes cannot be mass

Position x

current time-step

previous time-step

time Right travelling wave Left travelling wave mid

time-step


conservative, though in practice small accumulating mass error may not be

problematic. The latter method (adjusting pressure waves to conserve mass) is not

desirable either, since this introduces non-physical distortions to the flow solution.

A useful compromise was found that yields good results for most situations. In this

approach mild pressure correction is applied (for any accumulating mass imbalance).

Initial cell pressure is calculated using both methods and but the mass based pressure

value is given heavier weight - in this case 3:1 was used. In this way changes to the

pressure waves are minimised but the scheme retains long term mass conservation.

The method is inevitably a compromise, and provision is made to for the user to

select the non-mass conservative solution if desired.

Step 2 (from above) is to calculate the mid cell pressure due to heat transfer and

chemical reactions. The ideal gas relation is used. Then the

change in magnitude of both left and right travelling waves is calculated as.

(4-9)

4.3.3 Re-meshing

The mesh spacing in a duct should ideally be such that the Courant number is unity.

Practically however, it is impossible to achieve this since any flow velocity will

result in differing wave speed for the left and right travelling waves. Moreover

changes in the temperature and specific heats will also change the wave speeds. Re-

meshing allows the mesh spacing of each duct in a model to be adjusted

independently from time to time to suit changing flow conditions. Re-meshing

replaces existing nodes and cells with new ones where the fluid and flow properties

are interpolated from the existing ones. Figure 4-7 illustrates.


Figure 4-7 Re-meshing a duct

Re-meshing inevitably introduces some interpolation errors in all flow and fluid

variables, although use of a high order monotonic fitted curve minimises this.

Nevertheless, unnecessary re-meshing is avoided through judicious choice of re-

meshing criteria. The details of the re-meshing criteria are given in Appendix X.

4.3.4 Supersonic flow

Though unlikely in an engine duct, supersonic duct flow is theoretically possible. In

this case the upstream travelling wave will be unable to propagate upstream. The

code will have to test for the presence of a travelling shock, as pictured in Figure 4-

8. Tracing the movement of the shock is not straight forward, though one solution is

to measure the mass in the cell and calculate the shock position to satisfy this. The

speed of the shock relative to the cell can be calculated using the normal shock

equations in Appendix XII.

Figure 4-8 Detection of a travelling shock

time

current time-step

previous time-step

position

Position

current time-step

previous time-step

Left travellingwave XL

L ft t lli

Right travellingwave XR

Super sonic flow

Sub sonic flow

Normal shock

flow


4.4 Flow Boundary solution

Equation (4-1) assumes constant area, constant property, frictionless, adiabatic flow.

These conditions are too restrictive to directly produce a useful model for the gas

flow in the ducts of real engines. The effects of area changes, friction and changes

in gas properties must be evaluated, and this is done at the mesh points along the

duct. Thus a duct is made up of a string of idealised ducts (cells) where equation (4-

1) and its derivations hold true. At the connection point of each of these idealised

segments, any area change, gas property change or friction is accounted for. The

duct boundaries are solved with the same theoretical treatment, thus a uniform

boundary treatment is used throughout.

It is convenient to name the waves according to whether they are travelling toward

the node (incident) or away (reflected). This is because ordinarily, the flow at a node

is established only by the values of the incident waves (unless the flow is super-

sonic)

Figure 4-9 Duct cell boundary nodes in space and time

In general, to allow for varying gas composition and temperature within a duct, the

gas properties on either side of a node will be different (in space and time). This is

illustrated in Figure 4-10.

Figure 4-10 Variation of gas properties around a node in space and time

Position

current time-step

previous time-step

time

Reflectedwaves

Incidentwaves

Position

time

Reflectedwaves Xr

Incident wavesXi

a Ra a0a b Rb

a0b

c Rc a0c

d Rd a0d


The flow area may change from one cell to the next. This will occur between the

end cells of two connected ducts of different cross section, and it will also occur

between cells of a tapered duct (since the cell space is assumed to be constant area).

More complex area changes are discussed in the following section.

4.4.1 Flow Types

Considering all possible kinds of flow between duct sections and/or large volumes,1

there are about 31 distinct possibilities as shown pictorially in Figure 4-11. Constant

area duct flow is a special case of type ~1 where the change in area is zero. Friction

or reducing area may cause a choke point such as in type ~2. A sub-sonic diverging

flow will typically experience increased pressure loss due to flow separation. (~3) If

a diverging flow is sonic or super-sonic on the upstream side it will expand further

unless it is decelerated and compressed by a standing shock. (~4-6). A super-sonic

flow may pass entirely through a duct section (~7) unless it becomes choked, or a

shock travelling upstream against the flow passes through the section. In both cases

a travelling shock on the upstream side first slows the flow to a sub-sonic speed

before it passes through section (~8-9). If the flow passes a restricted area between

two ducts, the flow is in two stages. (~10-19). If a duct is connected to a volume

which is supplying the flow, the kinetic energy and pressure of the gas in the volume

are fixed. The inlet flow is assumed to be sub-sonic, though the flow may choke.

(~20-24) If the duct is supplying a flow to a volume, then the pressure on the

downstream side is fixed to the volume pressure (~25, ~28) unless the flow has

choked, in which case the sonic condition fixes the downstream pressure, and the

flow will expand somewhat as it enters the volume (~26, ~27, ~29). If two volumes

are connected directly by a short orifice, the flow may be sub-sonic or it may choke

(~30-31).

1 In the context of a gas dynamic model, a volume is a part of the model where gas dynamic effects are neglected, such as the atmosphere and cylinder. Volumes are typically larger in cross section than ducts, and are modelled as zero dimensional thermodynamic control volumes. A large reservoir such as the atmosphere can be modelled as an infinitely large volume which supplies a steady pressure to any connecting ducts.


Figure 4-11 Catalogue of all flow types considered

A correct solution for any one of the cases above is not especially difficult - the

difficulty lies in managing the complexity in providing for so many possibilities.

Increasing complexity runs the risk of inadvertent programming error or code

maintenance problems. It is beneficial to arrange the different flow types according

to families and use as much common code as possible.

4.4.2 Solution method overview

The general solution of the flow at boundaries is obtained as a quasi-steady, quasi

1D, calorically perfect flow. These conditions are discussed below.

~1

~2

~3

~4

~5

~6

~7

~8

~9

~10

~11

~12

~13

~14

~15

~16

~17

~18

~19

~20

~21

~22

~23

~24

~25

~26

~27

~28

~29

~30

~31

Single stage flow Two stage flow Volume flow

super-sonic

travelling shock

standing shock

sonic flow

converging flow

diverging flow

stagnant inlet

stagnant outlet

Flow is from left to right


Steady Flow - The flow at a node (be it a cell or duct boundary) conceptually passes

through an infinitesimal control volume. This allows it to be solved as a steady state

problem since there can be no accumulation of mass or energy within the (small)

control volume. The flow is quasi-steady. It is unclear if there is any alternative to

this approach. Chalet et al. [35] claim to avoid the assumption of steady flow, but

are referring instead to the application of a tuning ‘governor coefficient’ which is a

function of flow Mach number and not based on steady flow-bench data. The quasi

steady assumption remains in place.

1D Flow - The flow is assumed to have uniform properties across any given cross

section. The flow area may change, thus it is not truly 1D but quasi 1D. Clearly

boundary layers, free jets and recirculation zones violate the 1D assumption, but

experimental validation work such as [27, 110] show remarkably faithful simulation

results notwithstanding. The influence of detached flow can be modelled by adding

extra friction at these locations or by including an area coefficient. These methods

are imperfect since they tend not to reproduce the experimental data at all operating

conditions. This author has attempted to model the effect of recent flow history on

flow separation [56]. A modified form of the model is described below in section

5.3.

Calorically Perfect Flow – The quasi steady flow at each node is assumed to have

constant specific heats. Thus the gas is modelled as locally perfect (at each given

instant in time), though specific heats are allowed to vary in both space and time.

The assumption of local calorically perfect flow is not a serious impediment for

engine modelling since there are relatively low pressure ratios (and hence

temperature change) across flow restrictions.

Figure 4-12 shows a typical single stage flow boundary (type ~1). The gas

properties ( , ) in the cell spaces are in general different to the gas properties of

the boundary flow. Conceptually, contact surfaces exist immediately adjacent to the

flow boundary. The pressure and velocity on either side of the flow boundary are

usually different. Unless the flow is isentropic, the downstream isentropic reference

speed of sound increases slightly.


Figure 4-12 A typical flow boundary showing all flow properties

4.4.3 Example solution

The solution of the flow shown in Figure 4-12 is shown to illustrate the general

procedure. This flow has five unknowns, namely upstream and downstream and

, and downstream . Three equations can be written for the quasi steady flow in

Figure 4-12, between position 1 and 2 (1 2). These are conservation of energy –

equation (4-10), conservation of mass – equation (4-11) and flow energy dissipation,

here described by the change in isentropic reference speed of sound – equation

(4-14).

In addition, two equations can be written for the incident pressure waves as they

traverse the contact surface (a 1 and 2 b). These are equations (4-12) and (4-13).

Note that they are derived from equations (4-6) and (4-7) above.

Energy equation 1 2

(4-10)

Continuity equation 1 2

(4-11)

Wave equation a 1

(4-12)

Wave equation 2 b

a0a a

a0b b

X1 X2 a01 a02

u1 u2

A1 A2

Xia Xib

Unknown values in bold. Flow is from left to right


(4-13)

equation 1 2

(4-14)

where frictional dissipation may be a function of unknown variables

and the term in the denominator is the should be specified at the pressure of the

friction process (either or both) . Further information on the friction model

used in this thesis is given in section 5.1.

The above five equations form a system of non-linear algebraic equations in five

unknowns which can be solved simultaneously using the Newton-Raphson method

for simultaneous equations.

If the flow on the downstream side exceeds the local sonic velocity, the downstream

wave equation (4-13) must be replaced with the sonic flow equation:

(4-15)

In this case, no information from the downstream side influences the flow.

The derivations of equations (4-10) to (4-15) are shown in Appendix VII.

4.4.4 Separated flow

If the flow is diverging and subsonic (flows ~3, ~12, ~17, ~22 in Figure 4-11), then

it will typically suffer increased pressure loss due to flow separation. This results in

increased downstream thermal energy which is represented by the isentropic

reference speed of sound . In this case a flow separation dissipation term is

added to equation (4-14) so that it becomes

(4-16)

Where is the energy dissipated due to wall friction and is the energy

dissipated due to flow separation. A model for is required. Neglecting wall

friction, classical constant pressure flow can be achieved by setting


(4-17)

so that the kinetic energy lost from upstream to downstream is entirely converted to

thermal energy instead of pressure recovery. Setting represents isentropic

flow ( ). A model for flow separation has been developed which specifies

to a proportion of the constant pressure (equal pressure) value. See section 5.3

for details.

4.4.5 Two stage flow

In many cases where a duct is joined to another duct or volume, the flow path is

restricted to a smaller cross-section than the flow either side. In this case, the flow

first accelerates toward the restricted throat, and then decelerates into the larger duct

section downstream. To solve this flow two further unknowns must be solved –

namely the velocity and pressure at the throat, and . In this case two flow

equations are added – the energy and continuity equations for the extra stage in the

flow. Note that for simplicity the flow toward the throat is assumed to be isentropic,

while the divergent flow away from the throat may have friction and flow separation

losses applied.

4.4.6 Volume flows

A flow from a volume to a duct differs from duct flow because the cross section of

the upstream flow is undefined and the continuity equation cannot be applied.

However the pressure and kinetic energy of the upstream side are fixed by the

volume state. And so since the upstream and are known, the downstream flow

can be solved using only three equations – energy equation (4-10), the downstream

wave equation (4-13), and the equation (4-14). In many cases, flow from a

volume to a duct should be modelled with a somewhat restricted throat to account for

the vena contracta. In this case, the flow will be two-stage. Note that non-zero

velocity in the volume/reservoir is technically possible, such as modelling the

atmospheric ram air intake on a racing vehicle. In this case, the result is an increase

in dynamic pressure.

The flow from a duct to a volume is a similar problem, however, it is now the

downstream side with fixed pressure and zero velocity. The unknown variables are

the upstream and , and the downstream . The equations necessary are the


energy equation (4-10), the upstream wave equation(4-12), and the equation (4-

14). A separated jet entering a large volume is generally expected to have no

pressure recovery ( ), so the energy dissipation specified in the equation

should be set to (equation (4-17)). If the flow first contracts to a throat

before entering the volume then it will be two-stage. If the flow velocity exceeds the

local sonic velocity, then the downstream volume pressure no longer controls the

flow. The flow may choke (equation (4-15), or it may be fully supersonic,

depending on the upstream conditions.

4.4.7 Shocks

The effect of a standing shock within a diverging section is automatically satisfied

by the energy, continuity and downstream wave equation, but the possibility of a

fully supersonic outlet must be checked, in which case the downstream wave

equation must be replaced by the equation (4-14).

On the other hand, if the upstream flow is supersonic, checks must be made to see if

the flow will choke, or have a shock migrate upstream from the exit. If the upstream

flow becomes shocked, a simple correction can be made to the upstream incident

wave on each iteration of the main Newton-Raphson solution. This works quite

effectively, since the value of the incident wave changes only slightly when it

traverses such a shock.

4.4.8 Change of reference frame

Ducts with moving boundaries (and by implication, moving internal nodes) can be

modelled by changing the velocity reference frame of the node compared to the cell

reference frame. The change of reference frame modifies the flow velocity (relative

to the computational mesh nodes), but not the pressure. The result is modified wave

values

(4-18)

Where is the velocity of the node relative to the wave’s original reference

frame and positive velocity is the rightward direction. Thus a wave’s effective

magnitude is increased by a node moving to meet it, and decreased if the node is

moving with it. and are the cell space gas properties in which the wave is

travelling.


Once the flow at the node has been solved, the flow velocity can be converted

back to the cell reference frame simply by

(4-19)

where is the flow velocity relative to the node.

4.4.9 Numerical solution of the flow

Since there are many possible flow geometries (Figure 4-11), as well as sub-sonic

and supersonic versions of each, the first task of the flow solver is to determine

which category of flow the problem belongs to. The direction of a flow through a

given geometry is a crucial factor, so it is important to be able to predict the flow

direction. This is done by calculating the stagnation pressure on both sides of the

flow.

(4-20)

The inlet side will be the side with the highest stagnation pressure.

Once the flow geometry has been analysed and the appropriate equation set

specified, the flow can be solved. In some special cases, the set of equations can be

reduced to a single equation in a single unknown; however most flows require the

simultaneous solution of multiple non-linear equations. The Newton-Raphson

method for simultaneous equations is used here. It requires an ‘initial guess’ for all

of the unknowns, and the solution process is faster and more reliable if the initial

guess is near the solution.

A simple method for obtaining an initial guess of a flow is to assume constant

pressure and constant density across the flow. Then the continuity equation becomes

and for the case of uniform gas properties

Combining these with equations (4-3) and (4-4) and re-arranging gives

(4-21)

and


(4-22)

where and are the incident waves as shown in Figure 4-10 and Figure 4-12.

Similar approximate solutions can be written for a flow to or from a volume. If there

is a restricted area (throat) between positions 1 and 2, then the velocity at the throat

can be approximated as

If the velocity at the narrowest cross-section exceeds the local sonic velocity, then it

can be clamped to the sonic velocity by solving for the choked flow as

Clearly the ‘constant pressure’ assumption is a significant approximation on real

compressible flows, though it can be quite close for the case of separated diffusing

flows.

It should be noted that an alternative method for obtaining the initial ‘guess’ is to use

the flow solution from the immediately preceding time step. Though this method

has not been used in this thesis, it has the advantage of being computationally

efficient. Nevertheless, some ‘back up’ system is necessary for cases where the

previous timestep flow is very different (such as reversed) or unavailable (such as an

opening valve).

***

Finally, solution of the full set of equations by the Newton-Raphson method is as

follows [98].

Consider the following system of non-linear equations

Then write


(4-23)

where the partial derivatives and the functions are evaluated at the current

approximation . Equation (4-23) is a system of linear equations with

unknowns which are the correction terms. After solving equation

(4-23), the next approximate solution is.

Equation (4-23) is solved using a built in matrix division function in Matlab [77].

Several iterations of the procedure are required to achieve a sufficiently converged

solution. The convergence criteria used for the flow calculation was for the residual

value of each equation to be below a certain magnitude. These were chosen by

considering the relative magnitudes of typical terms in each equation.


4.5 Mass and thermal energy transport

The flow of mass and thermal energy can be tracked through the duct system. This

is necessary if any significant variation in gas properties is expected, or if the flow of

certain chemical species is of interest. In engines, it is important to model the

variations in exhaust gas temperature, and this especially the case for two stroke

engines with short circuit flow introducing pulses of low temperature air into the

exhaust duct. In this code conservation variables (mass and isentropic reference

temperature ) are stored in each cell space. This is not the only way to track mass

and heat. An alternative implementation could store only boundary node data.

4.5.1 Cell mass and temperature

Each cell is a control volume with two mass flows and as sketched in Figure

4-13.

Figure 4-13 Mass and thermal transport

If the positive mass flow is defined as flow into the cell, then the cell mass at a new

(mid) timestep can be calculated by assuming the average flow rates since the last

timestep are and .

(4-24)

is the timestep. If the model includes several gas species, then this calculation is

done species by species.

The new cell isentropic reference temperature is approximated as

Position

current time-step

previous time-step

mid time-step


(4-25)

where the specific heats of the mass flows are assumed constant and identical to the

cell specific heats. Alternatively for true energy conservation, a more rigorous

calculation requiring evaluation of internal energy at would have.

(4-26)

The reference temperature of the new cell could then be back calculated based on

the final cell internal energy . Note that in this thesis, the less rigorous equation

(4-25) was used.

The isentropic reference temperature conveniently uncouples pressure wave energy

from thermal energy, since it is independent of pressure. Foley et al [50] show that

uncoupling wave energy from gas properties avoids cumulative temperature errors.

4.5.2 Boundary flow properties

There are three fluid properties necessary to define an ideal gas for unsteady gas

dynamics - specific heats and , and isentropic reference temperature . Most

other properties of relevance can be derived from these. In this work, the fluid

properties tracked through the model are species mass fraction and reference

temperature . The specific heats are then calculated based on the resulting mixture

composition and temperature.

One simple method to specify the properties of the flow across a cell boundary is to

set them equal to the upstream cell values. This method results in a diffusive flow,

where initially sharp changes in fluid properties are quickly smeared. Some

smearing is physically legitimate since engine duct flow is typically turbulent,

however, the simulated rate of diffusion may be greatly exaggerated.

A model that is capable of preserving discontinuities has been developed bu this

author. It involves constructing a poly-line across the cell upstream of each boundary

flow in each fluid property as illustrated in Figure 4-14. The values of the flow

property at the previous timestep at the upstream and downstream boundary are

and . The cell average of the flow property at the mid time step instant is .

One of two families of curves are used, depending on whether the mid cell value is

closer to the upstream or downstream boundary values. The location of the knee


in the curve is determined by the criteria that the area under the curve matches that

of the mid cell value . The boundary flow value at the current timestep is then

found as the average value of the curve between and , where is determined

by the estimated flow velocity at the current timestep. Note that the dotted lines on

the lower part of Figure 4-14 represent particle path lines.

Figure 4-14 Calculating boundary flow properties

This solution preserves sharp changes on fluid properties. Extra diffusion can be

added to the solution by moving the point further upstream. A mixing co-

efficient was defined as a fraction of the cell length (0-100%) and this fractional

length was subtracted from the zero mixing position. This method produces

more-or-less consistent mixing effect for both high and low speed flows. In the case

of high speed flows, the time step needs to be small enough to prevent the value

exceeding about 25% of the cell length. Figure 4-15 shows the effect of different

mixing coefficients on an initial temperature discontinuity.

Position

current time-step

previous time-step

mid time-step

0

Flui

d pr

oper

ty

Approximate flow pathline

Flow direction

0


Figure 4-15 Temperature transport with different mixing coefficients

300

320

340

360

380

400

-10 -5 0 5 10

300

320

340

360

380

400

-10 -5 0 5 10cells

Nodal temperatures after initial discontinuity traversed 20 cells. Flow from left to right

Curves show results for different mixing coefficients

T (K)

T (K)

0 0.1

0.5 1

0 0.1

0.5 1

Flow 300m/s

Flow 30m/s


4.6 Validation using analytical results

4.6.1 Shock tube problem

The shock tube problem is a useful validation tool for gas dynamics codes because it

contains a shock wave, expansion wave and a contact discontinuity and also has an

exact 1 solution. It presents a severe test of the code’s ability to handle and preserve

sharp discontinuities. Comparisons of various gas dynamics codes using the shock

tube problem can be found in [42, 124].

Table 4-1 Shock tube setup Left Side Right side

(bar) 5 1 (K) 1200 300 (m/s) 0 0 (J/kg/K) 287 287

1.4 1.4 Shock tube length=1m

Diaphragm location=0.5m Computational mesh spacing 0.01m

Time step 0.01ms Results at t=0.5ms

Table 4-1 shows the initial conditions and other settings for the shock tube that was

simulated using the gas dynamics model. Friction and heat transfer were set to zero.

Note that all the shock tube plots below show the nodal (cell boundary) values.

Figure 4-16 shows the results of the model without explicit mass conservation. The

shock front location is well predicted and is resolved over about 2.5 cell spaces and

no overshoot is visible. The expansion wave velocity is slightly over-predicted and

is slightly smeared. The post shock values are all slightly wrong due to the inbuilt

assumption that pressure waves will cause isentropic compression or expansion (as

opposed to the compression process in a shock which is non-isentropic. The

temperature discontinuity is sharply defined and is testament to the transport

property model’s ability to retain sharp discontinuities. (see section 4.5.2 above).

Figure 4-17 shows the pressure and velocity for the same problem but with mass

conservation enforced using the method described above in section 4.5.1. As

expected, this causes some numerical disturbance to the solution with a slight

overshoot apparent around the shock and also just downstream of the expansion

wave peak. The enforced mass conservation is not overly disruptive to the solution,

and interestingly appears to somewhat improve the post shock pressure and velocity.

1 The solution is iterative but converges towards the exact value


Figure 4-16 Standard shock tube results

Figure 4-18 shows a comparison between the second order wave interpolation

method used in this model, and the linear interpolation method that is commonly

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pressure (bar)

0

100

200

300

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Velocity (m/s)

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Temperature (K)

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Density (kg/m3)

shock tube position (m)

Mass conservation off

exact model

exact model

exact model

exact model


used in wave action based codes such as the mesh MOC. The additional smearing of

the first order method is apparent.

It is noteworthy that the shock position is well predicted since no explicit shock

handling routine is used. However, it should be remarked that in any case, this test is

not especially representative of the typical flow in engine ducts, where the pressure

ratios are much lower and shocks usually do not have time to fully develop. In this

context the failure to properly model the non-isentropic compression process across

a shock is unimportant. Nevertheless the code in its present form would be

unsuitable to model problems with very strong shock propagation.

Figure 4-17 Shock tube with mass conservation

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pressure (bar)

-100

0

100

200

300

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Velocity (m/s) exact model

exact model




Figure 4-18 Shock tube comparison between first and second order wave

interpolation

4.6.2 Fanno / Rayleigh flow

Two cases of steady flow were compared to exact solutions. Steady flow with

friction and no heat transfer – Fanno flow – was tested by setting a duct with a high

pressure reservoir to the left and a low pressure reservoir to the right. The resulting

simulation was run for a duct with 10 cells till steady flow. The flow at the duct

entry is initially subsonic and high pressure, and as it progresses down the duct it

loses pressure and accelerates until it chokes at the duct outlet. The analytical

solution for Fanno flow based on the downstream boundary flow is plotted for

comparison in Figure 4-19. The results from the gas dynamics model correspond

very well.

Steady flow with heat transfer – Rayleigh flow – was tested in a similar manner. In

this case, friction is set to zero and heat is extracted from each cell at a rate of

6000J/kg/s. The duct has 10 cells. The flow leaves the high pressure upstream

reservoir and enters the duct in a slightly choked state. As heat is drawn from the

flow the pressure increases and the velocity decreases. The analytical solution for

Rayleigh flow based on the downstream duct boundary flow is plotted for

comparison in Figure 4-20. The model drifts slightly from the exact solution but the

error is relatively small. This good result is a significant validation of the heat

transfer implementation (section 4.5.1) since the change in pressure and velocity of

the flow is brought about solely by equal the modification (decrease) of the right and

left travelling waves at each cell at each time step.

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4

Exact

1st order

2nd order

0.5

1

1.5

2

2.5

3

0.72 0.77 0.82

Pressure (bar)




Figure 4-19 Fanno flow

Figure 4-20 Rayleigh flow

The exact solutions for Fanno and Rayleigh flow are shown in Appendix XIII.

320

330

340

350

360

370

1

1.2

1.4

1.6

1.8

2

2.2

0.5 0.6 0.7 0.8 0.9 1

P (bar) T (K)

Exact

P model

T model

1020

1040

1060

1080

1100

0.6

0.7

0.8

0.9

1

0.650.70.750.80.850.90.951

P (bar) T (K)

Exact

P model

T model

Upstream pressure 2.6bar, upstream temperature 394.17K Downstream pressure 1bar

Air, R=287J/kg-K, =1.4 Mass conservation on

0.7 0.8Mach

number

Upstream pressure 1.277bar, upstream temperature 1286.8K Downstream pressure 1bar

Air, R=287J/kg-K, =1.4 Mass conservation on

Mach number


4.6.3 Numerical smearing

To test the amount of numerical smearing, a high frequency triangular pressure pulse

with a wavelength of 12 cells is introduced into a straight, frictionless, adiabatic duct

and propagates through the duct for 100 mesh spaces. The amplitude of the wave is

small enough that wave distortion due to uneven propagation velocity is negligible.

Figure 4-21 shows the performance of the second order interpolation model

compared to the first order method for different Courant numbers. Both methods

show accumulating interpolation error however there is significantly less smearing

for the second order method which preserves better detail. A phase error is apparent

in the second order model though it should be noted that this smear test pushes the

code’s limits in resolution, and waves with a longer wavelength propagate with

accurate speed.

Clearly, Courant numbers close to unity produce better results, but the first order

method fails spectacularly if the number exceeds one. Compared to linear

interpolation the second order method is markedly improved, with only a small

increase in computational effort.

Figure 4-21 Smearing of a triangular pulse traversing 100 mesh spaces

-90

-60

-30

0

30

60

gau

ge p

resu

re (P

a)

-90

-60

-30

0

30

60

90

gaug

e Pr

essu

re (P

a)

Wave direction 1.01 0.9

0.8 0.4

exact

0.6

Wave direction

1.01

0.9 0.8

0.4

exact

0.6

Cells

Cells

Curves show results for different Courant numbers

Second order wave interpolation With mass conservation

W di ti

First order wave interpolation


4.7 Summary

The gas dynamic model presented in this chapter has been developed with the

demanding requirements of 2-stroke engine modelling in mind (see section 4.1.2). It

is a wave based method, similar to the MOC, but with improved spatial resolution

due to second order pressure wave discretisation. It is fully ‘non-homentropic’, or

more precisely, allows variable gas properties and temperatures throughout. The

boundary flow solutions allow non-isentropic flows and are fully energy

conservative. The boundary flow solution is common to both duct boundaries and

internal cell boundaries.

Mass and thermal energy are tracked throughout the model, ensuring that gas

properties at various locations are accurately represented. A high resolution mass

and energy transport model allows the user to reliably specify degrees of mixing.

The model is capable of full mass conservation.

The basic gas dynamic model has been validated against a range of analytical test

cases, and shows good performance. The model is further validated against a range

of experimental results below in Chapter 7.

The model can serve as a fully capable basis for modelling internal combustion

engines. However it has not been fully validated against supersonic flows, and some

difficulties here are yet to be resolved. Thus, the model may not be presently

suitable for modelling cases such as supersonic wind tunnel transients or high speed

shock tubes and gun barrels. Similarly, it is not suitable for processes involving very

strong shock fronts, since the compression process is assumed isentropic.

Furthermore, the model assumes the ideal gas relation , so caution is

recommended before this model is applied to potentially non-ideal fluids, such as in

CNG pipe lines.

90

. . . .

91

Chapter 5 Other sub models

This chapter describes the various other sub models that have not yet been addressed

in chapters 3 and 4, but which are significant parts of the overall engine model.

Section 5.1 deals with duct friction and heat transfer together since they are related

phenomena. Even though 1D model allows friction work to be specified (see section

4.4.3) the value is modelled in the present section. Likewise, section 4.3.2 described

the method for heat transfer to be incorporated in the 1D model. The actual value of

the heat transfer must be modelled, and this is the purpose of the present section.

Section 5.2 describes various combustion cylinder sub-models. The Main

thermodynamic volume model described previously in section 3.1 requires inputs for

heat transfer, gas blowby, fuel injection rate and combustion rate. Models for all of

these processes are described here.

Section 5.3 gives the details of a separated flow model introduced previously in

section 4.4.4. Separated flow is a common occurrence in engine ducts, and its

treatment influences the results of the 1D flow model significantly (See section 7.6

below for examples of the significance of the separated flow model)

Section 5.4 deals with the issue of coefficients of flow. It explains the importance of

defining flow coefficients in terms of area, not mass flow. It also describes the way

flow coefficients were implemented in the engine model.

Section 5.5 Describes the multi body dynamics model. This model is necessary for

a free-piston engine, since the trajectories of the pistons and valves are not

prescribed by mechanical linkages.

Chapter 5 Other sub models 92

5.1 Duct friction and heat transfer

Friction and heat transfer in engine ducts are difficult to model due to the complex,

highly unsteady nature of the flows. They are analogous phenomena, both

dependant on the mode of the boundary layer (be it laminar or turbulent). Time

resolved measurements of heat transfer in ducts with unsteady gas flow [20, 130]

have shown that steady correlations are somewhat inaccurate when used to model

instantaneous heat transfer. Most significantly, steady correlations fail to capture the

elevated heat transfer during the turbulent decay period after passage of a flow pulse.

The same problem exists for modelling friction under unsteady conditions. Using

flow over a flat plate as an analogy (see Figure 5-1), the wall shear stress becomes a

function of both velocity and boundary layer development, with a discontinuity

between laminar and turbulent modes.

Figure 5-1 Coefficient of friction for flow over a flat plate [90]

Consider an initially stationary fluid being rapidly impelled by a pressure wave

down a duct for a few milliseconds, before becoming stationary again after the wave

passes. For some initial period of the impulsive flow, the developing flow will be

largely coherent and have a thin boundary layer. It is only after it has travelled some

distance down the duct that the boundary layer thickens and turbulence begins to

form. Conversely, once the flow pulse becomes stationary again, residual turbulent

kinetic energy in the fluid continues to transport heat energy to or from the near wall

area, enhancing heat transfer.

An accurate unsteady friction model would have to model the boundary layer

development of impulsive flow. However, since pipe wall friction is typically not a

0

0.001

0.002

0.003

0.004

0.005

1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

coef

ficie

nt o

f fric

tion

Rex

Turbulent smooth plate

Laminar

Transition


crucial parameter in engine models, there is little discussion in the engine modelling

literature on this issue.

Heat transfer in unsteady conditions is even more complex. Several phenomena are

at work. First, in impulsive flow heat transfer enhancing turbulence will take time

to develop and then once the flow slows, time to dissipate. Secondly, parts of a duct

immediately downstream of some flow restriction will have elevated turbulence

intensity. This turbulent kinetic energy will be convected downstream and will

simultaneously decay. Thirdly, rapid changes in pressure cause compression heating

or expansion cooling of the gas in the thermal boundary layer, and the effect of this

on heat transfer can be significant [20]

5.1.1 Friction model

Provision is made for duct friction in the unsteady gas dynamics model in equation

(4-14) where friction is specified as specific work. The physical situation is

sketched in Figure 5-2. A slug of fluid is passing a computational node and is

dissipating energy through friction, resulting in a flow-wise reduction in pressure

and a corresponding increase in the thermal energy of the fluid. At the instant in

time of the analysis, the slug is centred around the node and has a combined length

of half of the upstream cell and half of the downstream cell.

Figure 5-2 Fluid element experiencing friction

The work per kilogram of fluid passing the node is

fluid element

duct

node

node

node


where F is the force opposing the fluid motion. The wall shear stress can be

expressed as

where is the Fanning friction factor or drag force coefficient and is the fluid

velocity relative to the pipe wall. Thus

(5-1)

where is the duct cross sectional area and is the circumference or wetted

perimeter.

The question of an appropriate friction coefficient was introduced above. A

common correlation for steady flows is the so called Blasius formula for turbulent

flow in smooth pipes with [90]

where the Reynolds number is

And is the hydraulic diameter

The viscosity of air-fuel mixture or exhaust gas can be estimated as a function of

temperature by a fitted power law given in [62]

(5-2)

However, given the intermittent nature of the flows in engine ducts, it is unlikely that

steady flow friction will hold. Based on this author’s analysis of experiments by

Kirkpatrick [68]it was found that a constant friction factor was most suitable as

(5-3)


5.1.2 Heat transfer

The application of heat transfer to the unsteady gas dynamics model is described in

section 4.3.2. The heat transfer is applied to the gas in a cell every (mid) timestep.

The physical situation is sketched in Figure 5-3. Heat is flowing across the walls of

the duct into or out of the gas flowing inside.

Figure 5-3 Fluid element experiencing heat transfer

The total heat transferred over one cell to the fluid in the duct over one timestep is

(5-4)

where is the timestep period. The heat transfer rate is estimated as

where is the fluid temperature, is the duct wall temperature, is the wetted

area and is the heat transfer coefficient.

According to the definition of the Nusselt number, the heat transfer coefficient is

related to the Nusselt number as

Where is the thermal conductivity of the fluid, is the Nusselt number and is

the characteristic length, usually the hydraulic diameter. The thermal conductivity of

the fluid is estimated based on an empirical expression given in [62]

(5-5)

fluid element

duct

node

node


5.1.3 Some other heat transfer models

Blair suggests calculating the instantaneous Nusselt number using the Reynolds

analogy of heat transfer [24] as.

Substituting in the Blasius friction co-efficient yields

(5-6)

Alternatively, substituting in the constant friction factor of equation (5-4) yields

(5-7)

A well-known heat transfer correlation by Dittus and Boelter [43] for steady

turbulent pipe flow is

where n is 0.33 for cooling and 0.4 for heating. Assuming a Prandtl number for the

gas in the duct of 0.77, the Dittus-Boelter correlation becomes approximately

(5-8)

Many researchers base a correlation on the mean duct flow [41, 62]. This method

has the disadvantage of requiring somewhat differing coefficients for differing

engines or duct geometries, and is not suitable for finding instantaneous heat

transfer. Zeng and Assanis [130] propose a two stage model for instantaneous heat

transfer coefficient. In the first stage of an impulsive flow, the model delays the

onset of elevated heat transfer by a correction term based on the instantaneous

acceleration of the flow. Once the flow decelerates to some critical speed, this

correlation no longer holds and a turbulence decay model is employed in this second

stage. Galindo et al [53] describe a model for heat transfer in exhaust ducting that

accounts for the delayed onset of elevated heat transfer during exhaust blowdown,

and also the proximity to the turbulence generating exhaust valve. The model uses a

modified Reynolds number whose velocity term is obtained as a weighted average of

previous calculation instants, and a linear coefficient which corrects for proximity to

the exhaust valve.


5.1.4 Heat transfer model

The model developed by this author is based on the concept of turbulent kinetic

energy in the flow. The model tracks a kinetic energy variable through the flow

and assumes a fixed large eddy length. Sources of turbulence are wall friction and

flow separation (see sections 4.4.3 and 4.4.4). Turbulent kinetic energy is only

monitored for the sake of evaluating heat transfer and does not enter into energy

calculations. The turbulent kinetic energy in a duct cell at each timestep is

calculated as

(5-9)

where the decay term is calculated according to the relationship

where is assumed equal to one and is the large eddy length. Assuming

isotropic turbulence, the turbulent kinetic energy is

then

(5-10)

The large eddy length is assumed to be proportional to the duct hydraulic diameter

The length scale coefficient determines the rate at which the turbulence decays. It

also indirectly controls the magnitude of the turbulence that is allowed to build up in

a duct. The turbulence decay model of Zeng and Assanis [130] used a value of

. However, this causes overly rapid decay of the strong turbulence

introduced downstream of flow restrictions, so the model here uses a much larger

large eddy length of around 20% of the hydraulic diameter.

At each node in the duct, turbulent kinetic energy is generated at the same rate as

friction heating. Thus for the flow at a node, the increase in turbulent kinetic energy

is proportional to the increase in reference temperature.


(5-11)

A correlation for heat transfer coefficient was developed as

(5-12)

where the turbulent Reynolds number is

and

Then the heat transfer coefficient is

(5-13)

This correlation was tuned to mimic the Dittus-Boelter correlation as far as possible

when run to steady state. It assumes that wall friction is calculated according to

equation (5-3) with , and the turbulent kinetic energy is calculated

according to equations (5-9), (5-10) and (5-11).

5.1.5 Some results of the heat transfer model

Figure 5-4 shows the heat transfer model compared to the Dittus-Boelter correlation

for a steady flow thorough a circular pipe. Curves show two different length scale

coefficients. Since the entrance is assumed to be lossless, the turbulence that drives

the heat transfer process takes several pipe diameters to build up to a steady level.

The smaller length scale coefficient (Cl=0.2) causes the model to produce

“developed flow” somewhat sooner than the larger length scale, however the values

converge after about 10 pipe diameters. The Dittus-Boelter correlation assumes

fully developed flow throughout.


Figure 5-4 Heat transfer model compared to Dittus-Boelter for steady flow

Figure 5-5 shows the turbulent kinetic energy calculated by the model for the flow

shown in Figure 5-4. The peak relative turbulence intensity is about 13% and 9.6%

for Cl=0.5 and Cl=0.2 respectively, which is far higher than the real case. This large

discrepancy is the result of the model assuming a constant large eddy length that is

much larger than typically used in modelling turbulence in steady pipe flow. The

large Cl value is used in this model to model the persistence of large, high-energy

turbulent structures caused by separated jet flow through restrictions such as valves.

Figure 5-5 Heat transfer model turbulent kinetic energy for different

turbulence length scales

Figure 5-6 shows the heat transfer for the case of a turbulence generating entrance.

An area coefficient of 0.75 is applied to the flow at the entrance to the pipe, and the

0

50

100

150

200

250

300

350

0 0.05 0.1 0.15 0.2 0.25 0.3

Ch (W

/K/m

2 )

m

Dittus-Boelter

Cl=0.5

Cl=0.2

0

100

200

300

400

0 0.05 0.1 0.15 0.2 0.25 0.3

ket (

J/gk

)

position (m)

Cl=0.5

Cl=0.2

Pipe diameter 25mm, velocity ~135m/s, gas temperature ~400K Flow is from left to right. Entrance assumed lossless



diverging flow downstream of the vena-contracta induces a small total pressure loss,

which is credited to increased turbulent energy. Unlike the ideal entry flow of

Figure 5-4, this case begins with above average heat transfer which decays to a

steady value after 8-10 diameters downstream. Unfortunately since the turbulence

generated at the pipe entry is constant, use of a larger length scale coefficient Cl

tends to reduce the effect of turbulence introduced by separated flows, compared to a

smaller choice of length scale.

Figure 5-7 shows the model compared to the Dittus-Boelter correlation for a slower

flow. Curves show results for two different length scale coefficients.

Figure 5-6 Heat transfer model for a turbulence generating inlet

Figure 5-7 Heat transfer model compared to Dittus-Boelter for low speed flow

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25 0.3

Ch (W

/K/m

2

position (m)

Dittus-Boelter

Cl=0.5

Cl=0.2

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

Ch (W

/K/m

2

position (m)

Dittus-Boelter

Cl=0.5

Cl=0.2

Pipe diameter 25mm, velocity ~125m/s, gas temperature ~400K Flow is from left to right. Entrance area coefficient 0.75

H t t f d l d t Ditt B lt f l



Figure 5-8 shows results of increasing the number of computational cells from 10 to

20. The results for this case are shown to be grid independent.

Figure 5-8 Heat transfer model for different cell spacing

The steady flow solution is largely unaffected by choice of large eddy length

(providing equations (5-10) and (5-12) both use the same length). Moreover, the

steady solution is not grid or timestep dependant.

To investigate the unsteady behaviour of the model, it was applied to a simulation of

a single shot experiment by Kirkpatrick[68]. Details of the experiment are given in

section 0. A flow pulse enters a pipe from the left and a pressure wave is transmitted

toward the right. Figure 5-9 shows the modelled heat transfer coefficient at two

instants as the pulse travels to the right. Intense turbulence generated by the flow

past the valve at the left enhances the heat transfer in this part of the pipe. The

turbulence is convected downstream but rapidly dissipates. Once the flow pulse

passes, decaying residual turbulence continues to enhance heat transfer. Choice of

length scale affects the rate at which it decays, but does not much effect the peak

heat transfer.

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

Ch (W

/K/m

2

position (m)

Dittus-Boelter

20 cells

10 cells

Pipe diameter 25mm, velocity ~20m/s, gas temperature ~400K Flow is from left to right. Cl=0.5 Entrance assumed lossless


Figure 5-9 Heat transfer modelled for single shot using different turbulence

length scales

Figure 5-10 shows the same case compared with a timestep of double the size. Note

that the mesh spacing is approximately doubled as well. The results are a close

match which demonstrates that the model is essentially not grid or timestep

dependent.

0

100

200

300

400

500

600

700

800

900

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6

Ch (W

/K/m

2

velo

city

(m/s

)

position (m)

velocity

Cl=0.2

Cl=0.5

-100

0

100

200

300

400

500

600

-20

0

20

40

60

80

100

120

0 1 2 3 4 5 6

Ch (W

/K/m

2

velo

city

(m/s

)

position (m)

velocity

Cl=0.2

Cl=0.5


Figure 5-10 Heat transfer modelled for single shot using different timestep size

5.1.6 Summary

A reliable, predictive heat transfer model for engine ducts is crucial for modelling

exhaust turbine performance (if one exists), for accurately predicting wave speeds in

engines with highly tuned exhausts, for temperature dependant exhaust after

treatment systems, and so forth. The model developed here is an attempt at a

universally applicable model. Its primary strength is in modelling the turbulence

decay period, which has been found to contribute around half of the total heat

transferred under conditions of intermittent flow [20]. Turbulence enhanced heat

transfer downstream of flow restrictions (ie in regions of separated flow) is naturally

incorporated into the model without having to resort to ad hoc correction terms such

as that described by [53].

The model does not consider the effect of pressure variation on boundary layer

temperature. Though this omission may not have a significant effect on total heat

transfer, it does appear to affect the instantaneous heat transfer rate [20].

Furthermore, the model does not appear to accurately model the delay of enhanced

heat transfer which has been noted by several experimental studies. Heat transfer in

the model is closely tied to flow velocity, at least during acceleration phase. The

model is, for better or worse, dependant on the friction model for turbulence

production. The failure of equation (5-3) to properly model friction for steady flow

pipe is surely cause for concern, but at this stage, a simple but faithful unsteady

friction model has not been developed. A subtler friction model may improve the

delay effect for heat transfer.

-100

0

100

200

300

400

500

600

-20

0

20

40

60

80

100

120

0 1 2 3 4 5 6

Ch (W

/K/m

2

velo

city

(m/s

)

position (m)

velocity

dt=0.0002

dt=0.0001


Other causes for concern are the assumption that all wall friction work is converted

to turbulent kinetic energy, and the arbitrary and unrealistic values for turbulence

that are produced for steady flow (see Figure 5-5). The model has only been

superficially validated ‘by eye’ against the results of Bauer et al [20] and Zeng and

Assanis [130], and also by matching single shot experiments by Kirkpatrick [68] (see

section 7.4). Further work would involve a more thorough theoretical analysis, and

further validations for a range of cases. An improved friction model may be helpful

in further improving the universality of the model.


5.2 Combustion cylinder models

5.2.1 Heat transfer

The instantaneous cylinder heat transfer is estimated using the Annand [14]

correlation where

Where a is constant of proportionality and is set to

as recommended by [49] for 2-stroke engines.

Defining the characteristic length as piston diameter gives

(5-14)

Viscosity is calculated as

(5-2)

Thermal conductivity is calculated as

(5-5)

Since the Pempek engine runs with a nearly constant speed, a mean piston speed of

8m/s was used as the characteristic velocity.

The wall temperature of the combustion chamber was assumed (no wall conduction

or water side heat transfer modelling was done). Three zones were designated -

cylinder head, cylinder wall and piston crown. The instantaneous heat transfer rate

was then estimated as

(5-15)

where T is the average gas temperature in the cylinder. Hot engine combustion

chamber temperatures were assumed as 400K


5.2.2 Blowby

Cylinder pressure measurements of the prototype engine suggested that significant

combustion chamber leakage was occurring (see section 2.3). The likely location of

this leakage was the piston-cylinder crevice, since it had no traditional piston rings.

Instead there was a cylinder mounted ring located far from the piston crown. A

sketch of the piston and cylinder layout is shown in Figure 5-11. The cylinder is

cylindrical, while the piston is slightly tapered to allow for greater thermal expansion

near the crown. The taper angle is 0.019 degrees, and the annular piston-cylinder

gap at the piston crown is 50 microns when cold. The cylinder bore is 68mm.

Figure 5-11 Sketch of piston-cylinder crevice

A CFD model of the problem was created by this author to analyse the likely blowby

flow characteristics. The model was 2D and used 9 cells across the long, narrow

crevice region. Details of the computational mesh are shown in Figure 5-12. The

model was for compressible flow, the density calculated according to the ideal gas

state equation, turbulence model was k- , and heat transfer was calculated based on

fixed wall temperatures. The cylinder pressure was simulated with a deforming

combustion chamber and a heat source term to simulate combustion heat release.

The wall velocities were modelled. The resulting cylinder pressure and crevice flow

are shown in Figure 5-13.

Cylinder wall

piston

labyrinth grooves cylinder ring


Figure 5-12 Blowby CFD model mesh

A simple blowby correlation was developed to allow computationally inexpensive

modelling of the blowby. The blowby correlation is

(5-16)

where

and

for

and

for

The correlation is compared to the CFD value is shown in Figure 5-13. All

quantities in the correlation are in SI units (but note the plot uses bar and

grams/second for convenience). To model the effect of a smaller or larger piston

cylinder gap, the correlation is scaled linearly.

cylinder piston

labyrinth grooves Piston-cylinder crevice


Figure 5-13 Blowby correlation

The model has limited validity. The behaviour in the case of a smaller gap has not

been modelled, and the crevice is assumed to be axisymmetric (i.e. a perfectly

aligned piston). In reality, the piston is probably eccentric, so the gap is much larger

on one side than the other. Furthermore, the piston axis may be tilted with respect to

the cylinder axis, introducing further three dimensional aspects to the problem.

More detailed modelling in three dimensions would be required to properly

understand the characteristics of the blowby flow here.

-20

0

20

40

60

80

100

0 0.005 0.01 0.015 0.02 0.025

bar,

g/s

seconds

P

mass flow CFD

mass flow correlation


5.2.3 Fuel injection

Fuel injection (even of liquid fuels) into the cylinder is assumed to be in the gaseous

state since the cylinder model assumes all species are ideal gases. Thus the process

being modelled is actually fuel evaporation. The rate, timing and total injected mass

are set by the user. The injection (or more accurately evaporation) is assumed to

happen at a constant rate, until the total injected mass specified by the user is

reached.

If the injected fuel is gaseous the molar specific enthalpy is calculated as

(5-17)

where is the enthalpy of formation of the gaseous phase at 298.15K and is

the sensible enthalpy of the gaseous phase. The subscript denotes the temperature

of the injected fuel.

If the fuel is initially liquid, the enthalpy of evaporation and the kinetic energy of the

jet must be accounted for.

(5-18)

where is the enthalpy of evaporation, is the sensible enthalpy of the liquid

fuel, and are the injection pressure and instantaneous cylinder pressure, and

is the liquid fuel density. The subscripts and denote liquid and gaseous

phase and is the boiling point of the fuel.

Figure 5-14 describes how the enthalpy of the liquid fuel is related to the gaseous

phase.

Figure 5-14 Finding fuel injection enthalpy

T

h

liquid phase

gaseous phase

298.15 Tinj

BP


5.2.4 Spark ignition combustion

The combustion rate for spark ignition is modelled with three parameters as shown

in Figure 5-15. The rise time determines the slope of the initial rate rise. After this

period of time, the rate is fixed at a constant value. The burn rate is also limited to a

proportion of the fuel mass still existing. Thus, the burn rate falls asymptotically to

zero as the unburned fuel in the cylinder is depleted.

Figure 5-15 Spark ignition combustion rate model

The parameters are:

Rise time

Maximum rate (normalised by cylinder mass)

Rate coefficient (multiplied by the remaining fuel mass)

time

burn rate kg/s

maximum rate

rise time

unburned fuel mass limited rate

ignition


5.2.5 Compression ignition combustion

Compression ignition is modelled using the fact that fuel-air mixing is finite rate.

The combustion rate is calculated by integrating the fuel mass in the cylinder WRT

time. The burn rate coefficient X is calculated at each time step as

(5-19)

Where is the previous value, is the fuel mass in the cylinder, is the

time step and is the fuel burned since the last time step. The combustion rate

is made proportional to X through use of a user-specified constant so that

(5-20)

The ignition timing is set by the user. The burn rate is related to the fuel injection

timing, as shown in Figure 5-16. Early injection increases the peak combustion rate.

The fuel injection period may overlap with the combustion period. After fuel

injection ceases, the combustion rate falls asymptotically to zero as the unburned

fuel in the cylinder is depleted.

Figure 5-16 Compression ignition combustion rate model

The model is crude, but allows the general combustion behaviour of direct injection

systems to be simulated.

time

burn rate kg/s

end of injection

Fuel mass limited rate

start of injection

ignition


5.3 Separated flow model

Perhaps the most difficult aspect of duct flow to model is the case where the flow

separates from the duct walls and generates energy dissipating turbulent eddies.

Unfortunately, this situation is relatively common in internal combustion engine

ducts. Separated (or diffusing) flow occurs at sudden enlargements in duct area,

turns, throttle valves, cylinder valves or ports and multi-pipe junctions. The problem

for modelling this kind of flow is to determine the level of pressure recovery from

the decelerating flow. It is well known that gently tapered diffusers can decelerate a

flow fairly efficiently, generating high pressure recovery, while steep diffusers are

inefficient, and the pressure recovery may be negligible. A common approach in

fluid mechanics to modelling pressure recovery in stepped ducts is to apply the

momentum equation to a control volume containing the area change, as illustrated in

Figure 5-17.

Figure 5-17 Control volume for applying the momentum equation to a diffusing

flow

Assuming that the pressure on the left hand boundary of the control volume is

uniformly equal to , and that the velocity across the cross section at either end of

the control volume is uniform, and neglecting wall friction, conservation of

momentum can be written as

(5-21)

This approximate solution is used by Blair [24] and has been shown to produce

generally good results [27, 68].

This author has developed an alternative model, based on direct specification of the

change in thermal energy of the flow, according to equation (4-16). (see section

4.4.3 and 4.4.4)

u2 2T02 / 2

A2

P1 Pu1 T01 / 1

A1

P2 Control volume


where is the energy dissipation due to wall friction, and is the energy

dissipation due to flow separation in a diffusing flow. The ability to control the

energy dissipation due to flow separation provides great flexibility. Many different

models can be suggested. It was observed that for a constant pressure flow

( ) the energy dissipation is equal to the loss of kinetic energy

(4-17)

Initially a function was developed that applied a fraction of to the flow

according to the relative deceleration of the flow [56]. Maximum deceleration was

for the case of an infinite expansion in duct area. This model conveniently allowed

both local and convective acceleration to be taken into account. The function is

given in equation (5-22)

(5-22)

However, it was found that the local acceleration term (rate of change of velocity at

a fixed point in the duct) was negligible compared to the convective acceleration

(acceleration of a particle due to its motion through the duct). Furthermore, the

model suffered from both time step and grid dependencies. The effect of

unsteadiness on duct flow separation is still not clear and further work is required

before the details of this complex phenomenon are understood.

In the meantime, a simpler function which does not take into account unsteady

effects and more closely mimics the momentum equation was implemented.

(5-23)

In the case of tapered ducts, equation (5-23) would be mesh dependant since the

ratio of duct areas approaches 1 when the mesh spacing approaches 0. The

correlation can be modified so that it becomes a function of taper slope only. A

sketch of one cell of a tapered duct is shown in Figure 5-18. The slope of the taper is

. If a ‘standard’ cell length is defined as , then the basic function (5-23)

becomes.

(4-16)


(5-24)

This equation is used for tapered ducts. The ‘standard’ cell length used in this thesis

was

Figure 5-18 Modified area ratio for tapered ducts

Equations (5-23) (for sudden area changes) and (5-24) (for tapered ducts) constitute

the separated flow model used in this thesis. Their performance is tested against

experimental results in section 7.6. They are grid and timestep independent, and are

easily applied to the numerical flow solution because they are effectively a constant

coefficient of .

duct centre line

L

l r1 r2 rstd


5.4 Flow area coefficient maps

It has been found that flow through sudden changes in area is often less than would

be predicted if the full cross section was utilised by the flow. One reason for this is

the vena contracta effect. Flow models account for this by including a flow

coefficient of some kind, which is tuned to reproduce experimental results.

Traditionally the flow coefficient or coefficient of discharge is defined as

(5-25)

where is the measured mass flow rate, and is the theoretically obtained

mass flow rate using the full geometric flow area and some form of isentropic nozzle

theory. However, this formulation is unsuitable for this gas dynamic model because

the model requires conservation of mass as a crucial part of the boundary flow model

(see section 4.4.3 above). What is needed therefore is not a mass flow coefficient,

but a flow area coefficient. This coefficient takes the form

(5-26)

where is the flow area which when used in the model reproduces the same

mass flow as the experiment. Determining the flow area coefficient requires an

iterative approach, where the flow model (including the all non-isentropic effects

such as flow separation) is used with successive guesses for until the

experimental pressure ratios and mass flow rate coincide [24]. With the area

coefficient thus defined, the flow area to be use in the model can be found by

multiplying the geometric flow area with the area coefficient.

In the absence of in-house flow data, this author analysed some published flow

coefficient data from QUB [25, 28, 29]. Unfortunately, this Author could not be

certain over the exact theoretical model used to derive the original coefficients, and

this uncertainty was compounded by some apparently inconsistent results.

Flow area coefficients for flow through an array of poppet valves was numerically

obtained using a 3D CFD model, then processing the data as described for equation

(5-26). Figure 5-19 shows the flow area coefficient map for flow through exhaust

valves (from a volume to a duct). The data points from the 3D CFD model are

shown by black circles. The green surface represents the map. The user can visually

modify the surface by dragging points of it up or down. This form for creating and


editing flow area coefficient maps was created using MatLab. Further details of the

area coefficient data structure can be found in Appendix VIII.

Figure 5-19 Example flow area coefficient map

Flow coefficients are a crucial part of a 1D gas dynamics model, as they allow

complex three dimensional flows to be accurately represented in a simplified 1D

world. This author has only built up a limited database of flow area coefficients, and

much of this data is still uncertain. Unfortunately, much existing published data is

difficult to utilise, either because its graphical presentation is not easy to digitise, or

because the exact details of the experiment are unavailable, or because the exact

details of the theoretical flow used to produce the coefficients is either not specified

or difficult to reproduce.


5.5 Multi body dynamics

The Pempek free-piston engine contains several un-constrained parts. These are the

free-piston (or mover), passive inlet valves and electromagnetic exhaust valves.

Figure 5-20 shows these parts in a simplified cutaway sketch.

Figure 5-20 Cutaway of FP3 showing moving parts

5.5.1 Mover dynamics

The forces on the mover are shown schematically in Figure 5-21.

Figure 5-21 Forces on the mover

The magnetic force is applied to the mover by the linear electric machine. This is

the only means of power extraction and also functions as a control on mover motion.

The generator load force is modelled using the same control algorithm as employed

on real engine (see equations (2-1) and (2-2).

Generator force is limited to a certain magnitude that corresponds to the electric

current limit imposed on the machine.

mover passive inlet valves exhaust valves

magnetic force

gas pressure friction

mover


Gas pressure exerts forces on all surfaces of the mover, but only force in the

direction of motion is considered.

Friction includes the mechanical friction of the magnet holder rubbing on the

generator stator surface, as well as electromagnetic effects such as eddy current drag

and magnetic hysteresis. Thus friction is modelled with a constant and variable

component

(5-27)

Where the friction force is opposite to the direction of relative motion, is a

constant, and is the relative motion between the mover and stator.

5.5.2 Exhaust valve dynamics

There were several different exhaust valve actuators used throughout the life of the

project. One of the versions is shown in Figure 5-22. It is made of a magnetically

permeable armature which is connected to an array of four poppet valves with small

springs. The springs ensured that all valves could seat.

Figure 5-22 Forces on the exhaust valves

valves

gas pressure

armature connecting

springs


For the sake of simplicity, the motion of the armature was prescribed (based on

experimental measurements) and the kinetic trajectory of the exhaust valves was

modelled with a combination of gas pressure force, spring force and friction. The

valves could possibly bounce when returning to the seated condition.

Friction from the valve guide was modelled as a constant friction force.

Gas pressure force was modelled by utilising an empirically derived valve force area

coefficient which was a function of both valve lift and pressure ratio as shown in

Figure 5-23. In these maps, the reference area is the circle inscribed by the valve

diameter. AR is the ratio of outer valve curtain area to reference area (or

4*lift/diameter). The data points were taken from a detailed 3D CFD model of the

exhaust valves and inlet valves undergoing steady flow, since physical data on the

aerodynamic forces on the valves was not available.

Figure 5-23 Aero force coefficients for normal flow through exhaust and inlet

valves

5.5.3 Passive inlet valve dynamics

A section view of the piston crown is shown in Figure 5-24, along with the passive

inlet valves (in this diagram there are four).

Normal Exhaust flow Normal inlet flow


Figure 5-24 Forces on the passive inlet valves

The passive inlet valves act under the combined influence of gas pressure forces,

closing spring forces and guide friction. At the same time, the piston on which they

are mounted is often rapidly accelerating. Like the exhaust valves, they may bounce

against their seats.

Gas pressure force was modelled in the same way as for the exhaust valves, using

the aerodynamic force coefficient maps shown in Figure 5-23.

The advanced inertia driven passive inlet valve design sketched above in Figure 2-12

contains a counterweigh mechanism. Although this mechanism has not yet been

modelled or built, the multi-body dynamics code described in the next section is

capable of modelling this kind of complex multi-body dynamic interaction.

5.5.4 Multi-body dynamics model

Bodies in the engine model represent moving parts in the real engine. A body may

be fixed (for example, the engine block), or it may be given a prescribed motion (for

example, a crank driven piston, or cam driven valve.) Alternatively, a body may

move with a kinetic trajectory in response to an array of forces acting on it. The

forces are of two categories. The first category is body interaction forces. These

include spring type forces, friction and impacts or seating forces. All of these forces

act equally and oppositely on a pair of bodies. The second category is any force

which affects only one specific body. Gas pressure force is one such force.

The instantaneous acceleration of a body is

valves

gas pressure

closing springs


(5-28)

The instantaneous velocity is found by assuming acceleration changes linearly

between the previous time step and the current

(5-29)

The instantaneous position is then

(5-30)

where is the time step increment and the subscript ‘prev’ denotes the preceding

time step value. Since some forces acting on a body may themselves depend on the

current position and velocity, the position and velocity are updated (iterated) several

times for each time step increment. The body must be sufficiently massive that its

motion over the period of one timestep is relatively ‘smooth’.

Impacts of one body against another must therefore be dealt with specially. If a

collision condition has been set by the user, and the trajectories of the two bodies are

found to have intersected, then the following procedure is followed:

Calculate the time of impact

Calculate the pre and post impact velocities and impact position

Update any friction forces to account for the reversed relative velocities.

Calculate the final positions and velocities

If the bodies are still intersecting at the end of the time step period, evaluate

them as a single lumped mass

The collision must satisfy conservation of momentum, and a coefficient of restitution

set by the user which specifies the ratio of the relative approach and rebound

velocities. If two bodies become pressed together (such as a seated poppet valve),

then they are treated as a lumped mass. A typical impact involving two bodies with

different mass is shown in Figure 5-25.


Figure 5-25 Typical collision trajectory

This one dimensional multi-body dynamics model allows many of the significant

moving parts of a free-piston engine to be modelled. The carefully implemented

collision module ensures that unconstrained poppet valve motion is accurately

simulated. The dynamics model can safely use the same time step size as the

cylinder and gas dynamics models which are typically set to give between 100 and

400 time steps per piston cycle. Some details of the data structures for bodies and

body interactions can be found in Appendix VIII

Time of impact

time

position

Final time

Initial time

body A

body B

123

Chapter 6 The engine model – integrating all sub

models

This chapter describes the integration of the sub-models detailed in Chapter 3,

Chapter 4 and Chapter 5. The model structure has been deliberately designed to be

multi-purpose. It is capable of modelling devices as simple as a single shock tube, to

as complicated as a multi-cylinder engine. In Chapter 7 below, this multi-purpose

model is applied first to modelling a series of single shot gas dynamics experiments

(by others), and secondly, to modelling the Pempek engine.

In any reciprocating internal combustion engine, there are one or more cylinders.

The flow of air, fuel and combustion products into and out of the cylinder is

controlled by various valves or ports. Usually a duct is interposed between the

cylinder and the outside world. In some cases, there are other volumes, such as

intake plenums, positive displacement compressors (such as crank case scavenged

engines), and mufflers. The application of the various sub models to the various

parts of an IC engine is shown diagrammatically in Figure 6-1

Figure 6-1 Integration of sub models to make an engine model

Thermodynamic control volume model (3.1)

Cylinder process models(5.2) -Heat transfer -Blowby -fuel injection rate -Combustion rate

(spark ignition or compression ignition)

Gas mixture properties model (3.2)

Reacting mixture model (3.3) (chemical equilibrium)

Heat transfer and friction Model (5.1)

ID gas dynamics model (Chapter 4)

Flow coefficient library (5.4)

Body dynamics model (5.5) -piston -exhaust valve inlet valve

separated flow model (5.3)

Engine ducts cylinder

Chapter 6 The engine model – integrating all sub models 124

6.1 Overview of model building blocks

The engine model is composed of three main building blocks. These are:

Ducts – the gas dynamic parts of the model. They are given a length and cross

section, which may be tapered. A duct is made of n cells with n+1 cell boundaries or

nodes. Each end of the duct may be connected to another duct or volume. If the end

is unconnected, the boundary is closed.

Volumes – the parts of the model where gas dynamic effects are neglected, such as

the atmosphere and cylinder. Volumes are typically larger in cross section than

ducts, and are modelled as zero dimensional thermodynamic control volumes. A

large reservoir such as the atmosphere can be modelled as an infinitely large volume

which supplies a steady pressure to any connecting ducts.

Bodies – the moving parts of the model are represented by bodies. These may be

parts such as pistons and valves. The Pempek free-piston engine had several floating

bodies which interacted through spring forces, contact and friction, such as the

mover and piston mounted passive inlet valves. Bodies can be specified in the

model to constrain the length of certain ducts, determine the volume of certain

volumes (such as the cylinders), or specify the geometric flow area through valves or

cylinder ports.

The interconnection of the various ducts and volumes must be specified, so also any

interaction between various bodies. The three main building blocks (Ducts,

Volumes and Bodies), along with the databases of interconnecting relationships are

described in more detail in Appendix VIII.


6.2 Calculation sequence for a complete time-step evaluation

The entire engine model operates on a common time step increment. The model

advances one time step at a time, calculating current conditions based on the

immediately preceding conditions. Since most of the pressures, flow rates, forces

and positions are interdependent, they are evaluated several times each time step in

an iterative fashion. The cylinder pressure is an especially important part of an

engine model. See Table 3-1 for the details of the volume conditions calculation.

The following list details the actual calculation sequence that was carried out for

each time step that the engine model advanced.

Shift time series data one place the right – at the beginning of a simulation, data

arrays are pre allocated with a user specified number of time series elements. Each

time the simulation advances one time step, the oldest data spills out of the array and

is lost. The current time step data occupies the lowest element (in MatLab element

1)

Calculate Bodies (1) – the first iteration simply extrapolates the velocity and

position base on the previous acceleration value. Any collisions are detected and

solved for.

Update all Duct nodal positions and velocities – this only affects ducts whose

lengths are defined by moving bodies. The initially estimated body positions are

assumed to be sufficiently accurate for these purposes.

Evaluate User Action function – this gives opportunity for special actions to be

performed. For instance the free-piston engine model models the cylinders as ducts

during the lower half of their strokes (when the gas exchange occurs). The User

Action function manages the transition from duct to volume when the mover crosses

the halfway point.

Evaluate Duct Cell properties – these are calculated from previous nodal flow

rates, flow temperature and, heat transfer.

Re-mesh the duct (if required) – test for the re-meshing criteria (see Appendix X)

Evaluate incident pressure waves – The pressure waves are advanced through

space. This calculation monitors for supersonic conditions (where a pressure wave

would be unable to propagate upstream). It also modifies the pressure wave values

for heat transfer. Mass conservation is also tested. Either cell mass is modified, or

pressure wave values are modified.


Calculate Volumes (1) – this first iteration of the volume calculation makes an

initial estimate of temperature by extrapolating the previous temperature rate of

change, and finding P using the ideal gas equation. The current temperature rate of

change can then be estimated based on the currently estimated volume rate of change

and previous values for flow rate and heat transfer. Current temperature and

pressure are then updated.

Evaluate flow connections (1) – with current volume pressure and temperatures

already estimated, current instantaneous flow to and from volumes is evaluated. If a

flow area coefficient must be retrieved from a map, the pressure ratio (PR) used is

averaged with the previous value, to prevent possible oscillatory situations. Flow

between ducts is also evaluated, in case the pressure here influences the motion of a

body (such a valve or piston).

Calculate Bodies (2) – this second iteration of the body motion calculation uses all

of the current values of spring force, friction, gas pressure to calculate current

instantaneous acceleration, and updates current velocity and position accordingly.

Calculate volumes (2) – volume heat transfer, chemical reactions and gas properties

are evaluated. With current flows, volume and volume rate of change known, the

current temperature rate of change is updated, allowing current temperature and

pressure to be evaluated.

Evaluate flow connections (2) – flow connections are evaluated a second time, with

updated volume conditions. Flow area coefficients are retrieved carefully using an

averaged PR with the previous iteration.

Calculate volumes (3) – the third and final volume evaluation is exactly the same as

the second iteration.

Calculate Bodies (3) – the third and final body motion calculation is exactly the

same as the second iteration. It may be unnecessary, except in cases where the body

is very sensitive to changes in volume pressure (such as a free-piston mover at peak

cylinder pressure).

Evaluate duct flow – even though the flow at internal duct nodes is not dependent

on any other parts of the model (such as volumes or bodies) this step is done last to

take advantage of having the current duct boundary flows known. This is helpful if

pressure wave information is coming from outside the duct boundary due to a

Courant number greater than 1. See Appendix IX for more details about this special

problem.


6.3 Programing details of the engine model

The engine model incorporating all of the sub-models was programmed using

MATLAB. A virtual engine consisted of any number of volumes, ducts and bodies,

as well as a flow connection database and a body interaction database. Every model

also had access to standard databases for gas properties, chemical reaction constants,

fuel properties, flow and force coefficients and user customisable functions. Both

the engine specific data and the standard data was implemented using the MATLAB

structure data type. This was not only convenient for readability of the code, but

also had the flexibility of allowing arrays of different size within a single data

structure. Furthermore, run-time re-sizing of data arrays was possible. This could

happen if a duct was dynamically re-meshed during simulation execution.

To assist in creating engine models, a graphical user interface was created. See

Appendix XIV for some further details of the graphical user interface.

The numeric class used for all floating point values was double precision (64bit).

Integer values were used where appropriate. Evaluation of boundary flows and

chemical equilibrium required the evaluation of a matrix division. The built-in

MARLAB function mdivide was used for this. Vectorised operations were used

where possible to speed up execution. The occurrence of runtime re-sizing or arrays

was minimised. Where possible, arguments to functions or subroutines were limited

to scalars or vectors, to save duplicating un-necessarily large data arrays in memory.

Although not implemented in the code, the structure of the model lends itself to

multi-processor execution, as many calculations can be carried out concurrently.

In total approximately 15,000 lines of code and comments were written, made up of

about 30 separate sub programs and functions. While this author chose MATLAB

due to ease of programing and de-bugging, and the availability of graphical

capability, the simulation code could otherwise have been written in any number of

programing languages such as C or Fortran.

128

. . . .

129

Chapter 7 Validations using experimental results

This chapter describes the validation work on the completed multi-purpose engine

model. First, a relatively simple model of a single shot test rig was created and

validated against an extensive set of single shot experiments conducted at Queens

university Belfast by Kirkpatrick [68]. This rig was specifically designed for

validating 1D gas dynamics codes by giving pressure data from a single generated

flow pulse without complicated wave reflections clouding the data. This has the

advantage of allowing fundamental flow processes to be easily analysed one at a

time, and decreasing the uncertainty in the experimental data about exactly what real

flow process is responsible for certain parts of the pressure record. Validation of the

model against these experiments was very valuable, as it provided a clear indication

of the real-world accuracy of the gas dynamics code in processes similar to those

found in IC engines.

Secondly, a complete model of the Pempek engine was created. It included valve

and piston dynamics, and cylinder models. The simulation results of the model are

compared to experimental engine run data; namely cylinder and compressor

pressures, and piston and exhaust valve trajectories. Clear conclusions are harder to

make here, due in part to drifting pressure signals, and also the low number of

pressure transducers on the prototype engine. The model appears to match the

limited experimental data quite well, but detailed comparison of the model results is

not possible.

Chapter 7 Validations using experimental results 130

7.1 Description of the single shot tests

7.1.1 Experimental setup of single shot rig

The single shot rig Kirkpatrick used consists of a cylinder connected to a series of

long pipes, where a single flow pulse is generated by the momentary opening of a

slide valve mechanism. The cylinder has a volume of 912cm2. The pipe is made of

aluminium and the small pipe diameter is 25mm. The slide valve has a movable

plate with an identical 25mm hole, surrounded by a sealing O-ring to ensure perfect

sealing before and after the test. The slide valve is actuated by a pneumatic impact

cylinder. In the original experiments the trajectory of the slide valve for each

actuation was carefully measured, however apart from an example case, these results

were not published. Further details of the experimental setup can be found in

Kirkpatrick’s PhD thesis [68] and in SAE papers [26, 27, 69].

The experiments were designed to test gas dynamic codes under a range of

conditions. Firstly, a code has to simulate the flow through the rapidly opening and

closing valve. The wave propagation velocity, wave distortion, attenuation and

resulting wave reflection due to friction and heat transfer was tested with straight

pipe (constant area) tests. Pressure wave transmission and reflection from passing

through a change in gas density was also tested with a special straight pipe

incorporating a second slide valve. A range of area changes were also tested, to test

for accuracy in these situations. The dimensions in mm of each of the experimental

rigs are shown in Figure 7-1 to Figure 7-8.

Figure 7-1 Straight pipe

Cylinder

slide valve

317

3691

5901

P1 P2


Figure 7-2 Straight pipe with density discontinuity

Figure 7-3 Sudden contraction

Figure 7-4 Convergent taper

Cylinder

slide valve

317

3097

3401

P1 P2

3703

5913

mid pipe slide valve

P3

Cylinder

slide valve

108 420

2454

P1 P2

2763 3060

P3

5270

Cylinder

slide valve

108 420

2454

P1 P2

2775 2970

P3

5481 3279


Figure 7-5 Sudden enlargement

Figure 7-6 Divergent taper

Figure 7-7 Short megaphone

Figure 7-8 Long megaphone

Cylinder

slide valve

317 3097

P1 P2

3394 3703

P3

6049

Cylinder

slide valve

317 3097

P1 P2

3406 3601

P3

6268 3922

Cylinder

slide valve

1705 3417

P1

3537

=84.8mm

Cylinder

slide valve

1705 3417

P1

4017

=109mm


7.1.2 Data compilation

Unfortunately the original data was not available, so the data was manually digitised

from plots in the original publications. The resolution and clarity of the original

graphics varied, however in most cases sufficiently good data could be recovered.

To allow easy reference to the original publications, the corresponding QUB

publication and figure number are listed in Appendix XI.

7.1.3 General model settings

The simulations were run with a global timestep of 0.1ms, and used dynamic re-

meshing. The mesh size varies and ranges from about 3.3cm up to 8cm to maintain

Courant numbers below 1.2. Where air was used in the model the composition was

21 parts Oxygen, 79 parts Nitrogen, 1 part water vapour and 0.038 parts Carbon

Dioxide. The mixture properties were calculated from the JANAF thermochemical

tables [38] as described in section 3.2. Mass conservation was imposed using the

method described in section 4.3.2. Friction was calculated as described in section

5.1.1 using a constant friction factor of 0.003. Heat transfer was calculated using the

turbulence based model described in section 5.1.4.


7.2 Slide valve tests

The slide valve mechanism was designed to produce a flow pulse with similar

characteristics to an engine valve or port opening event, while ensuring that perfect

sealing of the cylinder was maintained before and after the event. Unfortunately, the

trajectory of the slide mechanism has been found to be somewhat variable, and since

the trajectory data for each shot was not published, this trajectory had to be guessed

and manually adjusted for each case. This undesirable unknown casts some

uncertainty over the performance of the model. Furthermore, the experimentally

derived flow area coefficients for this valve were unclear, so a map was estimated

based on a combination of cylinder to pipe and pipe to cylinder data, and matching

with single shot results. These flow area coefficient maps are illustrated in Figure 7-

9. It would be preferable to build a flow area coefficient map directly from

experimental data, since this manually adjusted map adds a layer of uncertainty in

the fidelity of the model results.

A series of results using the straight pipe as sketched in Figure 7-1 at pressure sensor

P1 are shown in Figure 7-10 to Figure 7-19. Initial conditions in the cylinder at

release are noted in under each figure, and unless otherwise stated is air. The pipe

for all tests has a wall temperature of 293K and initially contains gas at 293K at a

pressure of 1 bar. Unless otherwise stated, the pipe initially contains air.

Most of the test cases are reproduced well by the model, though the flow area

coefficient has a strong influence on the exact shape of the pulse. The peak

pressures for the cases with different cylinder gas properties are well matched.

The characteristics of the post pulse pressure are also telling of the model’s fidelity.

Friction generates a steady reflection of the pressure wave, which for positive pulses

results in elevated post pulse pressure, as can be seen in Figure 7-10 and Figure 7-13.

Pressure waves propagating into denser gas also generate continual positive

reflections and this can be clearly seen in the air into CO2 cases (Figure 7-11 and

Figure 7-14). Conversely, heat transfer to the pipe walls always results in a

reduction in pressure wave values (both right and left travelling waves), so this effect

lowers the post pulse pressure somewhat. This balancing of friction effect and heat

transfer effect is a powerful way to calibrate models for the relative strength of

friction and heat transfer. The magnitudes of friction and heat transfer can then be

gauged by the attenuation of a pressure pulse since both effects reduce the wave

strength see section 7.4 below. Pressure waves propagating into less dense gas also


generate a negative reflection, as can be seen in the post pulse pressure of Figure 7-

12 and Figure 7-15.

The high pressure, high temperature release of Figure 7-17 matches the experimental

data well. The flow through the valve in this case is choked for the entire duration of

the valve opening.

The suction pulses (or rarefaction waves) shown in Figure 7-18 and Figure 7-19 also

match the experimental data well, although again, the flow area coefficient map

strongly influences the exact shape of this curve. The post pulse pressure is also

fairly well predicted.

Figure 7-20 to Figure 7-22 show the results from the so called “short pipe” shots on

the divergent taper rig which is sketched in Figure 7-4. In this rig, a short 25mm

diameter pipe is connected directly to the slide valve and then transitions to a

80.2mm diameter pipe. This situation creates a much higher pressure ratio across the

slide valve due to the presence of the nearby flow enlargement. While the suction

pulses in Figure 7-20 and Figure 7-21 are reproduced fairly well, the positive pulse

in Figure 7-22 is not. An improvement can be made by setting the flow area

coefficient to unity. It may be appropriate to modify the flow area coefficient map

so that flows at high pressure ratio are not significantly restricted. Further

investigation is necessary to identify the cause of the error. This is an important

flow type to clarify, because it is very similar to exhaust blowdown in many engines,

where an exhaust passage transitions to a larger diameter exhaust pipe in close

proximity to the valves.

Figure 7-9 Flow area coefficients used for slide valve

Cylinder to pipe flow Pipe to cylinder flow


Figure 7-10 Slide valve Prel =1.5 bar, Trel=293K

Figure 7-11 Slide valve Prel =1.5 bar, Trel=293K, air in cylinder, CO2 in pipe

Figure 7-12 Slide valve Prel =1.5 bar, Trel=293K, CO2 in cylinder, air in pipe

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)

model

experiment

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)



Figure 7-14 Slide valve Prel =2.4 bar, Trel=293K, air in cylinder, CO2 in pipe

Figure 7-15 Slide valve Prel =2.4bar, Trel=293K, CO2 in cylinder, air in pipe

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)


Figure 7-16 Slide valve Prel =2 bar, Trel=623K

Figure 7-17 Slide valve Prel =5 bar, Trel=623K


0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.002 0.004 0.006 0.008 0.01 0.012

pres

sure

ratio

t (s)

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)



Figure 7-20 Slide valve Prel =0.5 bar, Trel=293K short pipe shot


0.85

0.9

0.95

1

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)



0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.002 0.004 0.006 0.008 0.01

pres

sure

ratio

t (s)

measured

with Ca

no Ca


7.3 P1 driven simulation

Although the simulation results for flow through the slide valve are broadly quite

good (as shown above in section 7.2), any deviation from experiment at this point is

carried on to pressure readings downstream, and it can then be difficult to determine

where subsequent deviations from experiment originate. Furthermore, individual

shots, even at the same release pressures, sometimes displayed varying pulse profiles

due to individual variations in slide valve trajectory.

To remove this layer of uncertainty from subsequent downstream measurements, and

to expedite the task of setting up simulations (since the exact slide valve trajectory

for each shot was unknown), the experimentally recorded pressure was used to

correct the pulse profile at P1. Importantly, the complete valve and cylinder was still

functioning in the model with all of the appropriate initial properties, and release

flows. This ensured that friction, heat transfer and density reflections were properly

preserved. After the completion of the initial release pulse period, the forced

correction at P1 was turned off, and the remainder of the simulation proceeded

normally. All of the simulation results for the single shot rig results shown in the

following sections 7.4, 7.5 and 7.6 were achieved using this method.

There are a few disadvantages with this strategy however, which should be noted

here. Firstly, it depends on reliable pressure readings at P1. Quite a few pressure

readings at P1 show clear signs of ringing, and some of these are probably due to the

sensor diaphragm being unable to respond rapidly enough to sudden changes in

pressure. Secondly, some of the “bumpiness” of the pressure signal is probably due

to large scale turbulent eddies moving past the wall sensor. By clamping the

pressure at P1 to the experimentally measured values, we inadvertently introduce

these potentially spurious signals into the simulation.


7.4 Straight pipe shots

Figure 7-23 to Figure 7-26 show the pressure at station P2 for the straight pipe rig

shown above in Figure 7-1. Initial conditions in the cylinder at release are noted in

under each figure. The gas is air. The pipe for all tests has a wall temperature of

293K and initially contained air at 293K at a pressure of 1 bar. The pipe diameter is

25mm.

The attenuation of the pressure wave by this location in the pipe is fairly well

predicted, as is the phasing of the positive waves. The suction waves however show

a marked phase error, revealing a small inaccuracy in wave speed prediction.

Figure 7-27 shows the pressure at station P3 in the straight pipe rig shown in Figure

7-2. This test used carbon dioxide in all parts of the rig. The simulation results over

this extended period show that the wave speed in carbon dioxide is well predicted, as

well as the multiple reflections, first off the closed downstream end of the pipe, then

off the closed slide valve at the upstream end of the pipe. Wave attenuation due to

friction and heat transfer is well predicted and results in a rising pressure between

the main pulses. The slight “bumpiness” visible in these parts of the pressure record

is a numerical side-effect of the enforced mass conservation method (as described in

section 4.3.2) which is problematic for steep fronted waves when the local Courant

number is greater than 1. This problem can be reduced by decreasing the Courant

number (either reducing the time step or increasing the mesh spacing), however, for

engine modelling applications, steep fronted waves are rare.

Figure 7-28 to Figure 7-35 show the results for a pressure wave traversing a density

discontinuity in the rig sketched in Figure 7-2. The series of tests labelled AAC

have air in the cylinder and air in the first pipe section, while the second pipe section

initially contains carbon dioxide. The series labelled CCA have carbon dioxide in

the cylinder and in the first pipe section, while the second pipe section initially

contains air.

The simulation results for pressure sensor P1 correspond exactly during the initial

pulse period because they are forced to (see section 7.3 above). However after the

initial pulse passes, the simulated pressure here is released and is genuinely the result

of simulation. Crucially, the expected reflection from the downstream density

discontinuity appears with correct phasing and reasonably accurate magnitude. The

simulated results display a sharp step in the reflected pulse, due to the sharpness of

the modelled gas discontinuity. The double step is due to nearby closed cylinder


slide valve reflecting the “reflection” back again past pressure sensor P1. It is

probable that in the real case, some mixing of air and carbon dioxide occurred at the

mid-pipe gas density interface in the few seconds between the opening of the

separating slide valve, and the arrival of the flow pulse from the upstream cylinder.

This would explain the more diffused reflection that is apparent in the experimental

results.

The simulation results for pressure sensor P3 show that the magnitude of the

transmitted pulse is correctly modelled in the simulation.

Figure 7-23 Straight pipe P2, Prel =0.5 bar, Trel=293K



0.70.75

0.80.85

0.90.95

11.05

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.860.88

0.90.920.940.960.98

1

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

1

1.05

1.1

1.15

1.2

1.25

1.3

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)



Figure 7-27 Density discontinuity P3, CCC, Prel =2.4 bar, Trel=293K, closed end

Figure 7-28 Density discontinuity P1, AAC, Prel =1.5 bar, Trel=293K


1

1.1

1.2

1.3

1.4

1.5

1.6

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

1

1.2

1.4

1.6

1.8

0.01 0.02 0.03 0.04 0.05 0.06

pres

sure

ratio

t (s)

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

1

1.1

1.2

1.3

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)




Figure 7-32 Density discontinuity P1, CCA, Prel =1.5 bar, Trel=293K


11.11.21.31.41.51.61.7

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

1

1.2

1.4

1.6

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.951

1.051.1

1.151.2

1.251.3

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

pres

sure

ratio

t (s)

1

1.1

1.2

1.3

0.012 0.014 0.016 0.018 0.02 0.022 0.024

pres

sure

ratio

t (s)




0.91

1.11.21.31.41.51.61.7

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

pres

sure

ratio

t (s)

1

1.2

1.4

1.6

0.012 0.014 0.016 0.018 0.02 0.022 0.024

pres

sure

ratio

t (s)


7.5 Converging flow

Figure 7-37 to Figure 7-44 show the results of a positive flow pulse traversing a

reduction in area. The sudden contraction rig is sketched in Figure 7-3, and the

convergent taper rig is sketched in Figure 7-4. The sudden contraction had a flow

area coefficient applied as shown in Figure 7-36. The small pipe diameter was

25mm and the large pipe diameter was 53mm or 80.2mm, and is listed below each

figure.

Figure 7-36 Flow area coefficients used for sudden area change

The results show that the flow behaviour for sudden contraction and gradual

contraction are very similar. The gradual taper is slightly more efficient at

transmitting a flow pulse. Simulation results match this well. Note that in the

model, converging flow is assumed to be isentropic (apart from some skin friction

and heat transfer). Also, the gradual taper has no area coefficient applied, the flow is

assumed to fill the entire cross section. Close inspection of the post pulse pressure

reveals significant ‘wigglyness’. This is caused by strong flow oscillation in the

short 108mm pipe attached immediately downstream of the slide valve, and is

reproduced quite well by the model. Note that the diverging flow model used in

these simulations was the model described in section 5.3.

From large section to small From small section to large


Figure 7-37 Sudden contraction 53mm P1, Prel =2.4bar, Trel=293K

Figure 7-38 Convergent taper 53mm P1, Prel =2.4bar, Trel=293K

Figure 7-39 Sudden contraction 53mm, P3, Prel =2.4bar, Trel=293K

Figure 7-40 Convergent taper 53mm, P3, Prel =2.4bar, Trel=293K

0.951

1.051.1

1.151.2

1.251.3

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.951

1.051.1

1.151.2

1.251.3

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.951

1.051.1

1.151.2

1.251.3

1.351.4

0.008 0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.951

1.051.1

1.151.2

1.251.3

1.351.4

0.008 0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)


Figure 7-41 Sudden contraction 80.2mm, P1, Prel =2.4bar, Trel=293K

Figure 7-42 Convergent taper 80.2mm, P1, Prel =2.4bar, Trel=293K

Figure 7-43 Sudden contraction 80.2mm, P3, Prel =2.4bar, Trel=293K

Figure 7-44 Convergent taper 80.2mm, P3, Prel =2.4bar, Trel=293K

0.95

1

1.05

1.1

1.15

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.95

1

1.05

1.1

1.15

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.95

1

1.05

1.1

1.15

1.2

1.25

0.008 0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.95

1

1.05

1.1

1.15

1.2

1.25

0.008 0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)


7.6 Diverging flow

Diverging flow presents the greatest challenge to 1D modelling, because it can

involve separated flow which is a complex phenomenon, depending on several

parameters such as geometry, flow speed and even possibly recent flow history. If

the flow is assumed isentropic, then all of the kinetic energy that is lost as a result of

the fluid slowing down is converted into increased pressure. At the other extreme,

the flow could be assumed “constant pressure” or more accurately equal pressure.

Blair [ref] recommends the use of the “momentum equation” to approximate

pressure recovery in tapered diffusing flows. A model which achieves a similar

approximation was developed and is described in section 5.3. In this thesis it is

called the pressure recovery model. In the following test cases three diffusing flow

models are tested – isentropic, pressure recovery, and equal pressure.

Figure 7-45 to Figure 7-60 show the results of a positive flow pulse traversing an

increase in area. The sudden enlargement rig is sketched in Figure 7-5 and the

divergent taper is sketched in Figure 7-6. The small pipe diameter was 25mm and the

large pipe diameter was 53mm or 80.2mm or 105.6mm, and is listed below each

figure. Note that no flow area coefficient is applied to these diverging flows.

The pressure record at sensor P1 shows the returning suction wave. In the case of

sudden enlargements, the equal pressure model is a good approximation. In the case

of gradual enlargements, the isentropic model is a good approximation. In both

cases the pressure recovery model is superior. The transmitted pressure pulse

recorded at sensor P3 shows that in most cases, the isentropic model gives the best

correspondence with the experiment. In fact all of the models under predict the peak

pressure of the transmitted pulse. A particularly puzzling result can be seen when

comparing signals at P1 and P3 for the sudden enlargement to 80.2mm at 1.5 bar

(Figure 7-51 and Figure 7-53). At P1, the isentropic model significantly over-

predicts the strength of the suction pulse, however at P3, the isentropic model

produces by far the best results. The cause of this result is unknown.

The small spike in the simulation results of the divergent taper in Figure 7-56 at

about 13.5ms is a numerical artefact of the model automatically re-meshing the

tapered portion of the duct. This issue does not affect straight sections, and could be

avoided by fixing the number of cells in tapered ducts.

Figure 7-57 and Figure 7-59 show the pressure at sensor P2 for the divergent taper.

These results are interesting because they show the reflected suction pulse partly


superimposed on the original pressure pulse. The pressure recovery model is clearly

superior at this position.

Figure 7-61 and Figure 7-62 show the results for a positive flow pulse traversing an

open ended divergent taper or “megaphone” – both a short, steep taper and a longer,

shallow taper. These flow rigs are sketched in Figure 7-7 and Figure 7-8

respectively. In the case of the short megaphone with an included angle of 280 the

equal pressure model give good results, while the isentropic model over-predicts the

strength of the suction pulse, suggesting that the flow is largely separated in this

case. In the case of the long megaphone with an included angle of 80 the isentropic

model gives good results suggesting that the flow remains largely attached in this

case. In both cases, the pressure recovery model also gives good results. The

secondary reflections off the closed upstream slide valve are also well predicted in

both phase and magnitude.

Figure 7-63 to Figure 7-68 show the results of a negative flow pulse traversing a

decrease in area. In this case the flow is toward the cylinder and therefore is also a

diffusing flow. In contrast to the case of positive flow pulse through an increasing

area, there is little difference between the three diffusing flow models.

Figure 7-69 and Figure 7-70 shows the sort and long megaphone tests with three

different timesteps (and mesh spacing). This is a test for mesh and time step

dependency. The model employed automatic dynamic re-meshing to maintain a

target Courant number of about 1 (maximum Courant number of 1.2). All

simulations used the pressure recovery model. The results of the simulation show

minimal mesh dependency. All but the coarsest of simulations retain good

resolution. Note that for the largest time step (0.4ms), the average cell length is

about 200mm, which is much courser than that typically used in 1D engine

modelling.


Figure 7-45 Sudden enlargement 53mm, P1, Prel =1.5bar, Trel=293K

Figure 7-46 Divergent taper 53mm, P1, Prel =1.5bar, Trel=293K

Figure 7-47 Sudden enlargement/ Divergent taper

53mm, P3, Prel =1.5bar, Trel=293K

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

experiment equal P P recovery isentropic

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

1

1.02

1.04

1.06

1.08

1.1

1.12

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

1

1.02

1.04

1.06

1.08

1.1

1.12

0.01 0.012 0.014 0.016 0.018 0.02t (s)


Figure 7-48 Sudden enlargement 53mm, P1, Prel =2.4bar, Trel=293K

Figure 7-49 Divergent taper 53mm, P1, Prel =2.4bar, Trel=293K



0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

1

1.05

1.1

1.15

1.2

1.25

1.3

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

1

1.05

1.1

1.15

1.2

1.25

1.3

0.01 0.012 0.014 0.016 0.018 0.02

t (s)


Figure 7-51 Sudden enlargement 80.2mm, P1, Prel =1.5bar, Trel=293K

Figure 7-52 Divergent taper 80.2mm, P1, Prel =1.5bar, Trel=293K


80.2mm, P3, Prel =1.5bar, Trel=293K

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.01 0.012 0.014 0.016 0.018

pres

sure

ratio

t (s)

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.01 0.012 0.014 0.016 0.018t (s)


Figure 7-54 Sudden enlargement 80.2mm, P1, Prel =2.4bar, Trel=293K



80.2mm, P3, Prel =2.4bar, Trel=293K

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.01 0.012 0.014 0.016 0.018 0.02

t (s)





0.6

0.8

1

1.2

1.4

1.6

0.007 0.009 0.011 0.013 0.015 0.017 0.019

pres

sure

ratio

t (s)

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.005 0.01 0.015 0.02 0.025 0.03

pres

sure

ratio

t (s)

0.6

0.8

1

1.2

1.4

1.6

0.007 0.009 0.011 0.013 0.015 0.017 0.019

pres

sure

ratio

t (s)



Figure 7-61 Short Megaphone P1, Prel =2.0bar, Trel=293K

Figure 7-62 Long Megaphone P1, Prel =2.0bar, Trel=293K

1

1.02

1.04

1.06

1.08

1.1

1.12

0.01 0.012 0.014 0.016 0.018 0.02

pres

sure

ratio

t (s)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

pres

sure

ratio

t (s)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

pres

sure

ratio

t (s)




Figure 7-65 Sudden contraction / Convergent taper


0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.75

0.8

0.85

0.9

0.95

1

0.008 0.01 0.012 0.014 0.016 0.018

pres

sure

ratio

t (s)

0.75

0.8

0.85

0.9

0.95

1

0.008 0.01 0.012 0.014 0.016 0.018t (s)




Figure 7-68 Sudden contraction / Convergent taper


0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.005 0.01 0.015 0.02 0.025

pres

sure

ratio

t (s)

0.88

0.9

0.92

0.94

0.96

0.98

1

0.008 0.01 0.012 0.014 0.016 0.018

pres

sure

ratio

t (s)

0.88

0.9

0.92

0.94

0.96

0.98

1

0.008 0.01 0.012 0.014 0.016 0.018t (s)


Figure 7-69 Short Megaphone P1, Prel =2.0bar, Trel=293K,

various simulation timesteps

Figure 7-70 Long Megaphone P1, Prel =2.0bar, Trel=293K,

various simulation timesteps

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.015 0.02 0.025 0.03 0.035 0.04

pres

sure

ratio

t (s)

measured

0.0004

0.0002

0.0001

0.00005

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.015 0.02 0.025 0.03 0.035 0.04

pres

sure

ratio

t (s)

measured

0.0004

0.0002

0.0001

0.00005


7.7 Modelling the Pempek engine

The complete engine model as described in chapters 3, 4,5 and 6 was developed with

its primary purpose being to model the Pempek free-piston engine. In this section,

the Pempek engine is modelled, with a view to check the model’s fidelity against

experimentally measured data (by Pempek). If the model is able to reproduce the

measured engine behaviour reliably, then it can also be used with reasonable

confidence to do predictive simulations.

One further benefit of such a model is that it can be used in conjunction with

physical engine testing to illuminate the detailed inner workings of the engine, in

particular gas dynamics and in the case of the Pempek engine valve motion.

Therefore this section closes with further model results which are instructive, even

though they aren’t validated against direct measurements.

7.7.1 Engine model setup

Figure 7-71shows the complex 3D shape of the inlet side ducting of the Pempek

engine. In actual fact, the compressor volume is even more complicated than the

simple prism shown. Also, the four exhaust valve passages (not shown) turn through

90 degrees and are junctioned together immediately downstream.

Figure 7-71 Pempek engine inlet side ducting half section 3D view

To create the 1D engine model, each part of the three dimensional engine ducting is

approximated as a straight duct section. Figure 7-72 shows the layout of the 1D

Pempek engine model. The ducting for only one cylinder is shown, since the inlet

and exhaust systems for each cylinder are completely independent. The leftmost

duct section is the atmospheric intake pipe which connects to the internal compressor

compressor volume

4 x passive inlet valves

4 x exhaust valves

Combustion cylinder


volume via a check valve. The rightmost duct is the end of the exhaust pipe and

terminates at the atmosphere. The length of each duct section is shown in mm above

the graphic. The length of the compressor volume and combustion cylinder is

dynamic and is determined by the instantaneous position of the mover. The cross

sectional area of each duct section is shown in mm2 below the graphic. Each duct

section is constant area.

Figure 7-72 Pempek engine1D model layout

The atmospheric temperature and pressure was 300K and 101325Pa. The

atmospheric composition was 21 parts Oxygen, 79 parts Nitrogen, 1 part water

vapour and 0.038 parts Carbon Dioxide. The chemical composition of the (diesel)

fuel was C10.8H18.7. Duct friction and heat transfer were modelled using the methods

described in section 5.1. The combustion cylinder was modelled as a duct during the

lower half of the piston motion. It was modelled as a thermodynamic control

volume during the high pressure part of the cycle. Cylinder heat transfer, blowby,

fuel injection and combustion were modelled using the methods described in section

5.2. The motion of the mover, inlet valves and exhaust valves were determined by

the equation of motion (5-28) and the multi-body dynamics model described in

section 5.5. The motion of the exhaust valve armature was prescribed based on

experimental data. The mass conservation option for 1D gas dynamics was turned

on. The model time step was 0.1ms. Dynamic re-meshing was employed.

The engine model was set up with the same control signals as the real engine. Data

from a single engine run was used for comparison against the model. The engine run

began with four un-fired cycles, followed by 14 fired cycles. The fourth unfired

100 180 190 80 160 105 60 60 42 100 76±x 56±x

combustion cylinder exhaust pipe

compressor volume

600

mm

mm2 5238 1000 728 1900

3421 1772 804 1500 1777 1521 1521

compressor check valve

passive inlet valves

exhaust valves


cycle and the fourth fired cycle are compared to the model, which was cycled several

times to achieve approximately steady operation.

7.7.2 Results compared to experiment

Results for the model compared to experiment for both motored and fired cycles are

shown in Figure 7-73 to Figure 7-78. Only the right cylinder is shown.

The experimental pressure measurements from the cylinder and compressor have an

unknown DC offset, so the absolute pressure must be guessed. Furthermore, some

short term drift in the pressure signal may be present. Mover position was inferred

from instantaneous generator coil voltages. See the footnotes on page 21 for more

information on the cylinder pressure and mover position instrumentation.

Figure 7-73 Motored engine comparison with experiment

-60

-40

-20

0

20

40

60

0

1

2

3

4

5

6

-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03

Posi

tion

(mm

)

P (b

ar)

t (s)

mover position

compressor pressure

cylinder pressure

model

experiment


Figure 7-74 Motored engine, indicator diagram

Figure 7-75 Motored engine, indicator diagram

0

10

20

30

40

50

60

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

P (b

ar)

position (mm)

model

experiment

0

1

2

3

4

5

6

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

P (b

ar)

position (mm)

model

experiment


Figure 7-76 Fired engine comparison with experiment

Figure 7-77 Fired engine, indicator diagram

-60

-40

-20

0

20

40

60

0

1

2

3

4

5

6

-0.005 0 0.005 0.01 0.015 0.02 0.025

Posi

tion

(mm

)

P (b

ar)

t (s)

mover position

cylinder pressure

compressor pressure

model

experiment

0

10

20

30

40

50

60

70

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

P (b

ar)

position (mm)

model

experiment


Figure 7-78 Fired engine, indicator diagram

The model mover trajectory is not a perfect match with the experiment. This is

probably due to velocity signal smoothing in the real engine control system causing a

somewhat different generator force profile. Notwithstanding the minor difference in

trajectory, the model predicts the stroke length and frequency quite well for both

motored and fired cases.

The pressure at the compressor volume is a useful point of comparison. It indirectly

reflects the flow of inlet air into the combustion cylinder through the passive inlet

valves. The position of the pressure sensor is at the extremity of the compressor

volume, so changes in pressure here lag behind changes in pressure near the inlet

valves. The model reproduces the compressor pressure quite well, considering that it

is modelled with a simplified arrangement of ducts. This good match with

experimental data suggests that the passive inlet valve flows are broadly accurate.

The cylinder pressure during the gas exchange period also matches the experimental

data reasonably well. This is further evidence that the modelled trajectory of the

passive inlet valves is broadly accurate. The cylinder pressure in the final stages of

the gas exchange period (just before the exhaust valves close is less well predicted,

suggesting that the exhaust gas dynamics contain some inaccuracies. This is not

surprising, since the gas dynamics in the exhaust pipe are quite complex. In the case

of the fired engine, exhaust blowdown generates a flow pulse, which when reflected

off the open end of the pipe, returns as a suction pulse to the exhaust valves. This

0

1

2

3

4

5

6

7

8

9

10

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

P (b

ar)

position (mm)

model

experiment


suction pulse reflects off the semi restricted valve area, and returns again to the open

pipe end. On reaching the pipe end, a weak pressure pulse returns to the cylinder,

reaching it just before the exhaust valves close, forcing a little gas back into the

cylinder. The final cylinder pressure is affected by the precise timing of the exhaust

pressure waves, the exhaust valve closing, as well as the way the inlet valve flow

contributes to the exhaust flow. Animation movies of the motored and fired

simulations are available on disk and are listed in Appendix XV.

Close examination of the indicator diagram for the motored case (Figure 7-74 and

Figure 7-75) shows that there is substantial loss of cylinder pressure during the

closed part of the cycle, resulting in a large area of negative work. Though part of

this effect is due to heat transfer, the largest contributor in this case is likely cylinder

leakage (blowby). Note that sensor drift may also contribute to this result. (See

footnote on page 21) Assuming the measurements are reasonably accurate, however,

the blowby rate estimated in the model appears to be too small.

7.7.3 More model results

The model gives access to details about engine operation that are not otherwise

available. This section gives a selection of modelled engine data to illustrate the

useful role of the engine model to interpret otherwise scanty experimental

measurements.

Figure 7-79 shows the simulated exhaust and inlet valve trajectories. It is interesting

to note the trajectory of the piston mounded passive inlet valves, as they respond to

the combined influence of gas pressure and a rapidly accelerating piston near BCD.

The figure also shows the mass flows through inlet and exhaust valves. It is

interesting to note the momentary reverse flow at the exhaust valve at around 19ms.


Figure 7-79 Valve trajectories and mass flows

Figure 7-80 shows the simulated mass of gas in the cylinder, and the cumulative total

of the supplied inlet air. According to the simulation, the delivery ratio is about 1.6,

which should produce good scavenging efficiency. It will also mean that a large

amount of cool, oxygen rich air enters the exhaust pipe after each cycle. The

simulated blowby results in a substantial loss of cylinder mass during the

compression stroke, and the early stages of combustion.

Figure 7-80 Cylinder mass and inlet mass

Figure 7-81 shows the simulated cylinder chemical composition. Note that nitrogen

is not shown on this plot. Fuel injection begins at about -1.5ms and combustion

begins at about 0.3ms. As combustion proceeds, the oxygen and fuel fractions fall,

while the carbon dioxide and water fractions increase. The mass of fuel that was

-0.025

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

0.275

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.005 0.01 0.015 0.02 0.025

mas

s flo

w ra

te (g

/ms)

posi

tion

(m)

t (s)

Exhaust valve armature (x10)

mover, inlet valve position

exhaust valve (x10) inlet valve

(x10)

exhaust mass flow rate

inlet mass flow rate

-0.06

0

0.06

0E+0

1E-4

2E-4

3E-4

4E-4

5E-4

-0.005 0 0.005 0.01 0.015 0.02 0.025

posi

tion

(m)

mas

s (kg

)

t (s)

cylinder mass

cumulative inlet mass



injected (15mg) results in a slightly rich mixture. Thus late in the combustion

process, once oxygen has been consumed, a small amount of carbon monoxide and

hydrogen is produced. Finally, during the gas exchange period, the spent

combustion gases are purged from the cylinder and replaced with a fresh air charge.

The model has about 8% residual exhaust remaining, though this result depends on

the mixing coefficient used for the cylinder flow, and can be adjusted by the user

(see section 4.5.2).

Figure 7-82 shows the simulated cylinder temperature and the gas mixture ratio of

frozen specific heats ( ). A large drop in happens with the introduction of the

fuel. The value of this gas property for air at room temperature is 1.4, but is

substantially lower for gas mixtures inside an engine under operating conditions.

See Appendix II for a short discussion on frozen specific heats.

Figure 7-81 Cylinder species mass fractions

Figure 7-82 Cylinder temperature and specific heat ratio

0%

5%

10%

15%

20%

-0.005 0 0.005 0.01 0.015 0.02

mas

s fra

ctio

n

t (s)

CO2 O2

H2O

CO

H2 fuel

1.2

1.25

1.3

1.35

1.4

0

250

500

750

1000

1250

1500

1750

2000

2250

-0.005 0 0.005 0.01 0.015 0.02

K

t (s)

cylinder temp

ratio of specific heats (frozen)

170

. . . .

171

Chapter 8 Predictive modelling

This chapter applies the multi-purpose engine model to two proposed design

modifications of the Pempek engine.

The existing engine had a high pressure compressor to ensure that sufficient air is

delivered to the cylinder. However, analysis of the compressor pressure shows that

it consumes almost 30% of the engine’s indicated work (section 2.4). Part 1 of this

chapter applies the gas dynamics engine model to the existing engine in order to

determine whether it is possible to lower the compressor pressure, but maintain

sufficient air delivery through the original passive inlet valves.

Part 2 of this chapter describes a possible radical re-design of the engine - back to a

more typical port inlet layout. Without the integrated compressor, alternative means

of driving the scavenging process must be provided. Part 3 examines the feasibility

of using exhaust pressure wave energy to drive the scavenging.

Chapter 8 Predictive modelling 172

8.1 Optimising the existing layout

A slightly modified version of the existing engine was designed and then tested

using the model. The clearance volume of the compressor was increased to lower

the compressor pressure (this could be achieved by installing pistons with more

internal cavity volume). To help the passive inlet valves open, the return spring

strength was reduced from 6.8N/mm to 0.5N/mm, and the closed position spring

compression increased from 0.1mm to 1mm. The exhaust valve opening period was

reduced from 10 ms to 7 ms. For simplicity, exhaust valve timing and injection

timing were fixed for all load levels. A tuned exhaust was added, which serves to

provide some suction effect during the scavenging period, and at high load, to

produce a plugging pulse after inlet valve closure, but just before exhaust valve

closure. The engine layout is shown in Figure 8-1.

Figure 8-1 Tuned exhaust pipe layout for existing engine

All other model settings are the same as described above in section 7.7.1. The

engine model was tested across the load range from 0-22 mg of injected fuel per

cycle and the results are given in Figure 8-2. As a result of the earlier closing

exhaust valve, the somewhat cooler inlet air (because of a lower compressor work),

and a slightly elevated final cylinder pressure, the mass of air trapped in the cylinder

was increased from around 280mg to 400mg. This allowed the maximum fuel

quantity to be increased from 14.5mg to 22mg. Since the modelled blowby was the

same as used for the existing engine, higher combustion pressure resulted in higher

blowby loss, which reduces indicated efficiency at higher load. The indicated work

for the modified compressor was 22 joules (compared to 48 joules for the existing

engine). The indicated efficiency is based on a fuel heating value of 43MJ/kg, and

the net indicated efficiency has the compressor work subtracted from the cylinder

work.

300 300 230 300 300 550

combustion cylinder

exhaust pipe

compressor volume


Figure 8-2 Results for optimised layout

Figure 8-3 shows the valve trajectories and mass flows for the engine at maximum

fuelling. The exhaust valves open slightly earlier than the existing engine (Figure 7-

79 above), however due to the lower compressor pressure, the passive inlet valves

open a little later than in the existing engine, and only have one main opening.

Figure 8-3 Valve trajectories and mass flows (full power)

Figure 8-4 shows the mass of gas in the cylinder, and the cumulative total of the

supplied inlet air. According to the simulation, the delivery ratio is about 1.2, which

should ensure reasonably effective scavenging. It also means that some cool,

oxygen rich gas mixture enters the exhaust pipe after each cycle. The simulated

blowby results in a substantial loss of cylinder mass during the compression stroke,

and the early stages of combustion.

0

100

200

300

400

500

600

0 5 10 15 20

mas

s (m

g)

mg fuel

air mass delivered

10%

20%

30%

40%

50%

0 5 10 15 20-50

050

100150200250300350

mg fuel

ener

gy (J

)

ind. energyind. efficiencynet ind. efficiency

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-0.07

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.07

0.01 0.015 0.02 0.025 0.03

mas

s flo

w ra

te (g

/ms)

Posi

tion

(m)

t (s)

exhaust valve armature (x10)


exhaust valve (x10) inlet valve (x10)




Figure 8-4 Cylinder mass and inlet mass (full power)

Figure 8-5 shows the cylinder chemical composition. Note that for clarity nitrogen

is not shown on this plot. Fuel injection begins at about 8.4 ms and combustion

begins at about 10.6 ms.

Figure 8-5 Cylinder species mass fractions (full power)

There is sufficient oxygen in the cylinder to burn 22mg of the diesel fuel. However

in actual fact, the maximum fuel rate would probably be less than this due to the

finite mixing rate of fuel spray. The maximum gas temperature with a full 22 mg of

fuel is about 2160 K, which may also produce unacceptably high NOx emissions.

Thus, on this measure too, maximum fuelling may have to be reduced somewhat.

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0E+0

1E-4

2E-4

3E-4

4E-4

5E-4

6E-4

0.005 0.01 0.015 0.02 0.025 0.03

posi

tion

(mm

)

mas

s (kg

)

t (s)

cylinder mass



0%

5%

10%

15%

20%

0.005 0.01 0.015 0.02 0.025 0.03

mas

s fra

ctio

n

t (s)

CO2 O2

H2O fuel


8.2 Port admission layout

The Pempek engine has an integral compressor and utilises passive inlet valves

mounted on the head of the piston. This novel mechanism takes away the need for

cylinder ports which can accelerate piston ring wear and increase oil consumption.

The overall engine package is extremely compact owing to the overlapped generator.

However, the passive inlet valves present some inherent difficulties. The trajectory

of the valves is not fixed and depends on both piston motion and pressure differential

across the valve. (both exhaust valve timing and compressor pressure). Also, long

term durability has not been demonstrated, and this is all the more questionable

given the repeated seating impacts, heat, and unlubricated location. Furthermore, the

passive inlet valves constrain the shape of the piston head compared to what would

otherwise be possible. The flow of inlet air through internal cavities of the piston

will result in heat transfer to the inlet charge which will reduce charging efficiency

and may increase NOx emissions.

A radical re-design of the Pempek engine using inlet ports is described in this

section. Figure 8-6 shows a cross section of the existing Pempek engine alongside a

modified design with cylinder inlet ports. Passive inlet valves, internal mover

cavities and compressor check valves are eliminated.

Figure 8-6 Modification of Pempek engine for port admission

Existing Pempek Engine

Modified engine with port admission


As with the original design, the mover is supported by the generator stator and the

piston does not directly touch the cylinder. Piston rings seal the small gap between

piston and cylinder and need some form of lubrication.

Of course, such a design modification takes away much of what makes the Pempek

engine unique (integral compressor and no cylinder ports). The issue of piston ring

lubrication and wear returns, and some form of external supercharging may be

necessary.

8.2.1 Contactless piston

The current Pempek engine experimented with an alternative cylinder sealing

method (as explained above in section 5.2.2) that used a cylinder ring at the bottom

of the cylinder liner. Although this resulted in unacceptably high cylinder leakage,

the blowby was not excessively large, and this result begs the question of whether

further refinement of a contactless ‘seal’ might not be possible.

A contactless piston would have to have a very precise fit to the cylinder, and be

accurately supported and aligned. The main mover bearings of the existing engine

consist of engineered plastic pads running on the large metal stator surface. (This

method was possible given the very low bearing pressure) However, a small bearing

clearance is necessary to ensure that the mover does not bind and to allow for some

thermal expansion of the mover. This clearance reduces the accuracy with which the

pistons can be aligned within the cylinder. Manufacturing tolerances of the complex

mover assembly, generator stator and cylinder liner further reduce piston alignment

accuracy.

One bearing technology that is capable of precise positioning is air bearings. For

example, the port admission layout shown above in Figure 8-6 could have an air

bearing pad at the bottom of the cylinder, with the piston being running surface. An

alternative design is shown in Figure 8-7 where the rear of the magnet holder

becomes the bearing surface. In both of these designs, and the cylinder structure

supports the mover. This frees the air gap part of the generator from being the

bearing surface, which is beneficial since neither the slotted stator surface nor the

surface mounted magnets on the mover are ideal bearing surfaces. Compressed air is

supplied to the air bearing pads.


Figure 8-7 Contactless pistons with air bearings

There is much that is doubtful about the designs suggested above. For instance, can

the pistons/mover be adequately cooled? Can the piston-cylinder gap be controlled

sufficiently (given differential thermal expansion) to maintain engine efficiency but

prevent catastrophic seizing? How closely, for that matter, can the air bearing

clearance be maintained? How strong might un-balanced pressure forced on the

piston sides be? How much parasitic power would be required to supply the air

bearings with compressed air?

Leaving aside for the time being the feasibility of such a contactless piston engine, it

is clear that such an engine could be almost maintenance free, and there would be no

need for lubricating oil.

8.2.2 External supercharging

The loss of the integral compressor means that some other method must be found to

charge the cylinder with fresh air every cycle. An electric supercharger to provide

low-pressure boost would be suitable. The constant speed nature of a free-piston

engine suggests that gas dynamic tuning could also be utilised to provide beneficial

air flow to the engine. Hibi and Ito [63] have proposed removing the exhaust

ventilation fan (that drove the cylinder charging process) and using in its place the

inertia effect of exhaust gas. The rest of this chapter applies the gas dynamics

engine model to a port admission version of the Pempek engine, in order to

investigate the engine performance over the full load range and to assess the viability

of using exhaust system gas dynamics to drive the scavenging process.

mover exhaust valves

inlet bearing surface

fresh air in

stator


8.3 Gas dynamics driven scavenging

8.3.1 Model setup

The same mover mass and piston bore and stroke of the Pempek engine were used

for the new port scavenged layout. Inlet port opening was determined by mover

position, and so was by definition symmetric around BDC. Many prototype free-

piston engines use exhaust ports too. Most are loop scavenged [7, 15, 75, 76, 114,

115, 125], though a couple are opposed piston uni-flow scavenged [63, 120]. In the

present model, a symmetrical, mover controlled exhaust opening was used so that

results here could also be applied to these other engines. Of course, more timing

flexibility would be available with poppet valve exhaust.

To simulate the flow restrictions due to an air filter, the intake ports were connected

to a large plenum (20 litres) with a restricted inlet from the atmosphere. To simulate

the flow exhaust flow restrictions such as a muffler and catalytic converter, the

exhaust system was connected to a large plenum (6 litres) with a restricted flow to

the atmosphere. The intake and exhaust plenum pressures are given in the modelling

results below, and represent the adverse pressure differential which must be

overcome to charge the engine by gas dynamics alone. Figure 8-8 shows the layout

of the port admission engine model. The length of each duct section is indicated

above the graphic, and the cross sectional area at each section is indicated below the

graphic. A short inlet passage leads from the inlet plenum (not shown) to the

cylinder. The exhaust pipe is made from a series of tapered ducts and terminates at

the exhaust plenum (not shown). Only one cylinder is shown, but the engine model

has two independent cylinders and exhaust systems.

Figure 8-8 Port admission engine model layout

The cylinder blowby was modelled at ¼ of the rate used to model the existing

engine. The exhaust pipe wall temperature was set at a constant 500 K. All other

model settings are the same as described above in section 7.7.1

850 300 400 260 480 400

cylinder exhaust pipe

200

1667 mm2 3421

1200 1100 1500 3000 6362 400 6362

mm


8.3.2 Results

The engine model was run with a range of fuelling levels, from 2.5mg of injected

fuel , up to 22.5 mg. The model typically took at least 4 cycles to settle to a steady

operating state after a change in fuel level. Figure 8-9 shows the general results for

the port admission layout. The engine relies on exhaust pressure each cycle to re-

charge the cylinder, so 2.5mg of fuel is barely enough to sustain sufficient air

delivery. Furthermore, pressure loss due to blowby and heat transfer becomes a

significant proportion at low fuel levels. However at 5mg of fuel and higher, the

engine is easily able to charge. Operating frequency increases with fuel. The slow

decline in indicated efficiency at higher fuelling levels is mainly attributable to the

increased specific heat of high temperature, combustion product rich mixture, though

increased ‘time loss’, blowby and heat transfer may also contribute.

Figure 8-9 General results for port admission layout

Figure 8-10 shows the port openings and mass flows for the engine at full power

setting. Figure 8-11 shows the cylinder pressure for the same cycle. The initial

rapid blowdown of the cylinder draws the cylinder pressure slightly below the intake

pressure. A suction condition at the exhaust port is maintained for about 3.5 ms after

which the inlet port closes. Soon after this, a so called ‘plugging pulse’ arrives at the

exhaust port and rapidly forces some of the recently exhausted gas back into the

cylinder, boosting the pressure to around 1 bar above atmospheric. Figure 8-12

shows modelled the inlet plenum and exhaust plenum pressures over one cycle.

Note that the plenums are connected to both cylinders.

Figure 8-13 shows the cylinder mass, oxygen mass and cumulative inlet flow.

0

100

200

300

400

500

600

0 5 10 15 20

mg

mg fuel

masstrapped

air massdelivered

0 5 10 15 200

5

10

15

20

25

30

35

40

mg fuel

Hz frequency

20%

30%

40%

50%

60%

0 5 10 15 200

50

100

150

200

250

300

350

400

mg fuel

J

ind. energy

ind. efficiency


Figure 8-10 Port openings and mass flows (full power)

Figure 8-11 Cylinder pressure during scavenging (full power)

Figure 8-12 Modelled inlet and exhaust plenum pressure (full power)

-0.07

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.07

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.01 0.0125 0.015 0.0175 0.02 0.0225 0.025 0.0275

Posi

tion

(m)

mas

s flo

w ra

te (g

/ms)

t (s)

exhaust port opening

mover position inlet port

opening



0.5

1

1.5

2

2.5

3

0.015 0.0175 0.02 0.0225 0.025

P (b

ar)

t (s)

cylinderinlet plenumexhaust plenum

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

0 0.005 0.01 0.015 0.02 0.025

P (b

ar)

t (s)

atmosphere

inlet plenum

exhaust plenum


Figure 8-13 Cylinder mass and inlet mass (full power)

Figure 8-14 shows a comparison of cycle temperatures for the range of fuelling

levels. Interestingly, the lowest fuelling level (2.5mg) actually had higher cylinder

temperatures due to low air delivery mass and high levels of residual exhaust gas.

Figure 8-14 Cycle temperatures

0E+00

1E-04

2E-04

3E-04

4E-04

5E-04

6E-04

0.015 0.0175 0.02 0.0225 0.025

mas

s (kg

)

t (s)

cylinder mass


O2 mass

250

500

750

1000

1250

1500

1750

2000

2250

0 0.02 0.04 0.06 0.08 0.1

T (K)

position (m)

22.5mg

20mg

15mg

175.mg

12.5mg

10mg

7.5mg 5mg

2.5mg


Figure 8-15 shows the cycle pressures for various fuelling levels. The lowest

fuelling level suffers from low cylinder mass due to the absence of any plugging

pulse. Figure 8-16 shows a comparison of cylinder pressure during scavenging. The

clear trend is for increasing strength of plugging pulse with increasing combustion

energy.

Figure 8-15 Cycle pressures

Figure 8-16 Cycle pressures during scavenging

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08 0.1

bar

position (m)

22.5mg

15mg

10mg

5mg

2.5mg

0.5

1

1.5

2

2.5

3

3.5

4

0.06 0.07 0.08 0.09 0.1 0.11

bar

position (m)

22.5mg

15mg

10mg

5mg

2.5mg

22.5mg

5mg


8.4 Conclusion

One of the Pempek engine’s distinguishing features is the passive inlet valves and

integrated compressor, which remove the need for problematic cylinder ports.

However, the current engine uses an enormous proportion of power to compress the

inlet air to over 2 bar above ambient. The results of the simulations in section 8.1

show that it quite possible to run the engine at lower compressor pressure. This re-

design reduces the compressor work from 48J to 22J, which is equivalent to a

pumping IMEP of 0.6bar. Furthermore, improvements in trapping efficiency

increase the maximum fuelling level from about 14mg to 22mg – translating to a

power increase of over 50%. Results of modelling an un-boosted engine (section

8.3) suggests that even lower compressor pressures are entirely possible, providing

that the passive inlet valves can be designed to operate without pressure across them.

However passive inlet valves, internal mover cavities and integral compressor are

somewhat complex and constrain the design of the mover and piston crown, heat

transfer to the inlet charge will be increased, and moreover, passive inlet valves are

un-proven for long term durability. The alternative of port inlet (and often exhaust)

is used by many free-piston engine prototypes, along with traditional piston rings.

One elegant possible alternative to piston rings is a completely contactless ‘seal’,

though the feasibility of this option is unknown.

Regardless of whether a contactless piston seal can be implemented, or if traditional

lubricated piston rings are used - the simulation results in section 8.3 show that a

tuned exhaust pipe can charge a two stroke engine without any added assistance

from external blowers. Moreover, this has been demonstrated for the case of port

inlet and port exhaust which is the most restrictive case.

Simulated free-piston engine operation was fairly robust, and was able to self-charge

from full power down to about 10%. The engine cannot self-charge when motoring,

so some method of purging the cylinders with air would be needed to start the

engine. One problem encountered was that a sudden increase in fuel load sometimes

triggered un-symmetric operation, where one cylinder was unable to fully charge.

Since stronger combustion in one cylinder resulted in higher trapped mass, this

produced a corresponding reduction in stroke length to TDC, which reduced the port

opening for the other cylinder (which is at BDC). This problem could be avoided by

increasing fuel level over two cycles instead of one. Alternatively, a less aggressive


plugging pulse may result in a more consistent trapped cylinder mass over the load

range (as opposed to the increasing trapped mass shown in Figure 8-9).

In the simulation, the entire exhaust pipe wall temperature was arbitrarily set at

500K. This is not realistic, of course, since the part of the pipe closest to the

cylinder will be significantly hotter than the downstream sections, and this will

depend on a range of factors like exhaust gas temperature and whether the pipe is

externally insulated. Since pressure wave speed is proportional to the square root of

gas temperature, modelling the complex interactions of continual wave pulsations in

the pipe depends on accurate gas temperature modelling. It may require some

experimental iterations with prototype pipes to refine the model for wall temperature

etc.

The tuned exhaust pipe which assists scavenging (in both cases analysed) is over 2m

long and may be difficult to package with the engine, though the pipe can be bent

around on itself without significant loss of performance (as is done, for example with

2-stroke motorcycles exhausts). The pipe could be made somewhat shorter (say 1m

shorter) if the strong plugging pulse was not needed. A weak plugging pulse would

still be available, but maximum engine power would be somewhat less (say 20%).

Alternatively, if the exhaust was controlled by poppet valves, then the exhaust valves

could close momentarily before the inlet port closed, preventing excessive loss of

cylinder mass to the exhaust. It may also be possibility is to have both cylinders

supplying a single tuned pipe exhaust.

The simulations that were done above paid no attention to intake system gas

dynamics, but it is likely that additional gains in charging efficiency can be made

here.

All this is to say, in the end, that a myriad of design options are available to take

advantage of a free-piston engine’s inherent constant speed nature, for improving or

controlling the charging of the engine. The complexity of wave interactions in

engine ducts mean that it is necessary to explicitly model the gas dynamics (‘rules of

thumb’ do not always work, and are sometimes misleading as Blair has pointed out

[24]). A 1D gas dynamics model such as the one developed here is an indispensable

tool for exploring design options for engine charging.

185

Chapter 9 Summary and conclusion

This thesis has covered a lot of territory, which can be conveniently grouped into

two main topics – namely free-piston engine technology developments and general

engine modelling. This wide subject range reflects the particular needs of the

Pempek project for a wide range of both results analysis and predictive modelling.

All free-piston engineer researchers develop some kind of model or models to test

their designs – indeed it is fairly easy to construct a basic free-piston engine cylinder

model to determine operating frequency and the like. Some of the models used by

others are very advanced, such as detailed in-cylinder CFD, detailed chemical

kinetics, and in some cases 1D gas dynamics programs.

The modelling tool that this author developed for the Pempek engine is

comparatively simple, as it does not attempt to model detailed in-cylinder processes,

however it lays great emphasis on accurately determining the inlet and exhaust flows

by modelling the gas dynamics of the entire inlet and exhaust duct. The model is

simple enough to be easy to set up, and fast to execute on a computer, which allows

effective design to be carried out on a large range of cylinder charging options. It is

also generally applicable to a wide range of other gas dynamic processes.

This author believes that too few free-piston researchers understand the potential of

gas dynamics to influence engine charging. (See section 1.5.4) Free-piston engines

(especially 2-stroke ones) are ideally suited to take advantage of gas dynamics

because they are essentially constant speed machines.

The following sections summarise the specific findings for the Pempek project, the

scope and usefulness of the model, the unique contributions made in the field of

modelling, and finally a few areas of suggested further research.

Chapter 9 Summary and conclusion 186

9.1 Findings for the Pempek project

This author’s association with Pempek Systems has lasted for about four years, and

during that time a range of engine performance issues were examined. Early work

included a simple free-piston engine cylinder model from which estimates of exhaust

blowdown pressure were used in the design of an updated exhaust valve actuator.

Other early work involved a detailed moving mesh CFD cylinder model for testing

the influence of inlet valve placement on cylinder scavenging. (see the movie

‘points’ in Appendix XV for an example). This work suggested that about 8%

residual exhaust would be retained, and that valve placement here did not greatly

affect scavenging efficiency, however exhaust pressure and inlet valve lift had to be

estimated. Following this work, the integral compressor was analysed and

recommended to be re-designed for lower pressure in order to reduce power

consumption and inlet air heating.

Findings that relate directly to the engine model described in this thesis are as

follows:

Passive inlet valve trajectory – the simulated valve trajectory showed that re

seating velocities were as high as 3.5m/s and that the valve typically opened at least

twice each cycle. Simulations showed that a weaker return spring could be used.

Lower compressor pressure possible – simulations showed that enough air

delivery could be made with a lower compressor pressure, especially if a tuned

exhaust pipe assisted in evacuating the cylinder after blowdown.

Exhaust tuning beneficial – simulations of a port admission version of the engine

showed that no compressor was necessary for cylinder charging if a carefully tuned

exhaust pipe was used. That is, according to modelling results, 2-stroke engines

operating in a narrow speed range can be charged by using the gas dynamic energy

of exhaust flows.


9.2 The model, its scope and usefulness

The engine model described in this thesis was developed to model the Pempek

engine. It is actually a collection of independent models for

thermodynamic control volumes

gas properties and chemical equilibrium

1D unsteady gas dynamic flow

1D multi body dynamics

various miscellaneous models such as heat transfer, combustion rate and

blowby

These models can be applied to a wide range of processes, not just internal

combustion engines. A graphical user interface was created to allow the user to

easily create, assemble and edit engine models. The multi body dynamics model is a

special requirement of free-piston engine modelling, and the correct handling of

collisions is particularly important for the Pempek engine with its passive inlet

valve; however other models for motion can be prescribed by the user.

Since it is important to easily visualise simulation data, two special graphing utilities

were created. One facilitates the plotting of time series data, while the other allows

gas dynamic animations, which give the engine designer clear insight into the gas

dynamic processes within the engine. Several of these animations are included with

this thesis and are listed in Appendix XV.

The 1D gas dynamics model allows variations in gas properties, is energy and mass

conservative, and includes an extensive treatment of boundary flows. It is ideally

suited to high performance two stroke engines with strong wave action, thermal and

gas property discontinuities and diverging and converging ducts. It provides for an

engine model which is comprehensive enough to the capture the significant physics

of the engine cycle (especially engine breathing), while being simple and fast enough

to allow rapid evaluation of design options such as

optimal valve timing

optimal valve sizing

optimal inlet and exhaust duct shape

necessary inlet charge pressurisation etc.

delivery ratio


9.3 Unique contributions to modelling art

During the course of developing the engine model, the author devised and

implemented several new methods.

Chemical equilibrium code – This code was based on the popular method of

equilibrium constants, however equation solution and the detailed code was written

entirely by the author. The solution is efficient and reliable, taking an average of 5

iterations of a 4x4 matrix division to reach sufficient accuracy. It can handle gas

mixtures without atoms of carbon, hydrogen or nitrogen - only oxygen is

compulsory. (See section 3.3)

Second order pressure wave interpolation method – Wave action gas dynamics

models are typically first order accurate since they use linear interpolation of

pressure wave values between mesh nodes. A simple but effective second order

interpolation method was devised and implemented. (See section 4.3.1) Smearing is

significantly reduced compared to the first order method, and Courant numbers

slightly above unity can be safely used.

Heat transfer and mass conservation in 1D model– The heat transfer

implementation is different to the original method from Queens University, Belfast.

It performs well in the Rayleigh flow test. Furthermore, the method includes a

provision for full mass conservation, which is notable, since wave action methods

usually are not fully mass conservative. (See section 4.3.2)

Duct re-meshing – This procedure allows the mesh spacing of individual ducts to be

changed mid simulation to better suit the prevailing flow conditions, and maintain a

Courant number closer to unity. (See section 4.3.3)

Moving mesh – this capability is useful for some situations where ducts deform,

such as a moving piston problem. This capability was added to the original method

by a simple change of reference frame for the travelling pressure waves (See section

4.4.8)

Fluid property interpolation method (with variable mixing) – The original

method used the upstream cell values for gas properties at nodal flow. This results

in unavoidable diffusion of thermal and mixture discontinuities. A reasonably

simple interpolation method was devised and implemented which allows the user to

set various rates of mixing and can maintain sharp gas property discontinuities. (See

section 4.5.2)


Duct friction and heat transfer rate model – This author’s analysis of

Kirkpatrick’s single shot data [68], as well as consideration of the process of

boundary layer development in impulsive type flows, found that a constant

coefficient of friction was more suitable than the more commonly used Blasius

formula.. (See section 5.1.1) Following similar reasoning, a heat transfer model was

devised which depended on turbulence kinetic energy in the bulk flow to account for

highly elevated heat transfer in instances of strong turbulence. (See section 5.1.4)

Diffusing flow model – The original gas dynamics method used the momentum

equation approximation to model diffusing flow pressure recovery. Initially, a

relationship which included flow unsteadiness [56] was developed, but in the

absence of showing a clear advantage, has been abandoned in favour of a simpler

steady relationship. (See section 5.3).


9.4 Contribution to free-piston engine research

This body of work, taken as a whole, offers two main contributions specific to the

field of free-piston research. The first is that it demonstration of the importance of

gas dynamics effects in two-stroke free-piston engines. Section 1.5.4 reviews the

existing references to gas dynamics modelling in recent free piston literature.

Further, details of recent free piston projects are given in Appendix I. These sections

show that not many free piston engine researchers have attempted to model or

exploit gas dynamic effects. Section 2.3, especially Figure 2-10 shows that the

Pempek engine with ordinary exhaust tube is clearly affected by gas dynamic effects

given the pressure fluctuations in the cylinder during charging of up to 0.5 bar. Even

in the un-optimised state this represents a fluctuation in cylinder trapped mass of +-

25%. The introduction to Chapter 4 demonstrates that from a purely theoretical

point of view, neglecting gas dynamics can lead to large errors in calculated mass

flows.

Conventional engine modellers have long been convinced of the power of gas

dynamics to help or harm engine performance. The modelling in this thesis of

possible design options for the Pempek engine (Chapter 8) shows that gas dynamics

effects have the power to dramatically improve the charging efficiency, and if

necessary, to charge the cylinder without assistance of intake blowers or

compressors. The wide range of possible design options that gas dynamic effects

introduces are discussed in the conclusion to Chapter 8 (section 8.4)

The second contribution to the field of free piston engine research is a thorough

literature survey of recent free piston projects (section 1.2) This survey is followed

by a discussion of some central free piston engine concerns such as piston motion

control (section 1.3). Finally, a more in depth report of the Pempek engine design

and experimental results (Chapter 2) allows those interested in the field to learn from

Pempek’s experience.


9.5 Further Work

During the course of the work, several avenues for further research were identified,

but because of time constraints were not pursued. With regard to unsteady 1D

modelling:

Multi-pipe junction model - A good multi-pipe junction model is an important part

of a fully capable engine model – though it was unnecessary for the existing Pempek

engine since the inlet and exhaust systems were independent for each cylinder.

There is quite a bit of literature on this topic, and perhaps understandably (given the

simplifications inherent in a 1D model), none of the models reported so far have

been able to reproduce experimental results for a full range of cases.

Diffusing flow model – This author believes there is more scope for investigating

this very typical flow mode. Evidence seems to point to reduced flow separation for

the case of impulsive flow compared to steady flows, but some reports are

conflicting. This flow mode is also important in multi-pipe junction flows, where

gas flows around quite sharp bends and experiences a loss of total pressure.

Unsteady duct heat transfer – A relatively simple model for unsteady heat transfer

based on modelled turbulence energy was quickly developed for this work. More

work needs to be done in examining the physical basis for such a model, and

validating and refining it against a range of experimental cases.

***

With regard to free-piston engines, it would be useful to do a study of the feasibility

of a contactless piston. (See section 8.2.1) Port inlet and/or exhaust has great

advantage over valve systems in terms of simplicity, zero power consumption and in

some configurations, uni-flow scavenging. Normally however, cylinder ports can

cause accelerated piston ring wear and increased oil consumption. An engine with a

ringless piston, no-contact ‘seal’ could be almost maintenance free. The free-piston

engine with its linear motion and low side forces is an ideal candidate for such a

technology. Probably the main difficulty with the concept is potential un-balanced

side pressure forces on the piston due to piston-cylinder eccentricity during the high

pressure part of the cycle. Thus, a careful piston dynamics study is required which

models the three dimensional piston-cylinder crevice flow and the resulting pressure

forces on the piston.

192

. . . .

193

PUBLICATIONS

Greg Gibbes and Guang Hong, Improving the accuracy of a 1D Gas Dynamics

Model, Proceedings of the 17th Australasia Fluid Mechanics Conference, Auckland,

New Zealand, 5-9 December 2010.

Greg Gibbes and Guang Hong, A General Boundary Solution Method for 1D Gas

Dynamic Models, Proceedings of the 17th Australasia Fluid Mechanics Conference,

Auckland, New Zealand, 5-9 December 2010.

Greg Gibbes and Guang Hong, A model for pressure recovery of flows in expanding

ducts for 1D gas dynamics simulations, Proceedings of the 15th Asia Pacific

Automotive Engineering Conference (APAC15), Hanoi, Vietnam, 26-28 October,

2009.

Greg P. Gibbes and Guang Hong, Numerical modelling the dynamics of a piston-

mounted passive inlet poppet valve, SAE 2007-32-0099, JSAE 20076599,

Proceedings of the 13th Small Engine Technology Conference (SETC), Niigata

Japan, 30 October-1 November 2007.

194

. . . .

195

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. . . .

207

APPENDICES

Appendix I Review of recent free-piston engine projects 209

Appendix II Specific heats of a reacting mixture 235

Appendix III Further applications of the energy equation 237

Appendix IV Tables of Thermodynamic Properties 241

Appendix V Method for Calculating Chemical Equilibrium of Combustion

Products 247

Appendix VI Derivation of fundamental one dimensional unsteady gas equation

255

Appendix VII Derivation of Boundary Flow Equations 259

Appendix VIII Model Data Structures 263

Appendix IX 2nd Order Interpolation – further details 269

Appendix X Re-meshing Criteria and Method 271

Appendix XI Single Shot Experiments Cross Reference 273

Appendix XII Derivation of normal shock equations 275

Appendix XIII Rayleigh and Fanno Flow 281

Appendix XIV Graphical user interface screen shots 285

Appendix XV Table of contents of data CD 291

208

. . . .

209

Appendix I

Review of recent free-piston engine projects

This review will focus on contemporary projects published from about the year 2000

onward. These recent projects are briefly summarised according to their supporting

institutions. Design details and results of particular relevance to this thesis are

emphasised. Thus more attention is paid to electric free-piston engine projects (as

opposed to hydraulic), projects with experimental results, and with similar

mechanical design issues to the engine studied in this project. A cursory summary of

linear electric machine projects is given, though mainly to provide context for the

current state of the field.

I-1 Overview of early Free-piston engines

The first commercially successful free-piston engines were designed in the 1920’s.

The early history of free-piston engines is well explored by Aichlmayr [8], and the

following details of early engines are summarised from his thesis. Other worthwhile

reviews can be found in [6] and [82].

Prolific inventor Raúl Pateras Pescara developed a number of designs for free-piston

air compressors beginning in 1922. At around the same time the German company

Junkers began developing similar air compressors, which were used to provide

compressed air for torpedo launch tubes on German submarines. Both the Pescara

and Junkers style machines were opposed piston designs utilising a synchronising

mechanism to ensure the pistons maintained perfect opposed motion. This design

was convenient for the purpose of gas exchange, since one piston uncovered exhaust

ports, while the other uncovered inlet ports. Air compression was generally multi

stage and was achieved with various integrated pistons. The Junkers design was

highly successful and following the Second World War captured Junkers free-piston

compressors were distributed to various U. S. firms, Universities, and laboratories

for evaluation and testing. Figure I-1 shows a cutaway view of a Junkers air

compressor.


Figure I-1 Junkers four-stage free-piston air compressor [8]

The next major phase of free-piston engine technology was free-piston gas

generators (gasifiers). These engines produced high pressure, high temperature gas

which was used to supply a gas turbine for power extraction. At a time when gas

turbine technology was in its infancy, a free-piston gasifier gave higher combustion

temperatures and pressures, while allowing the gas turbine to run on relatively low

temperature partially expanded gas. French company SIGMA produced the GS-34

in 1944 based on Pescara’s designs. This engine had a final power output of around

600kW and was produced for large scale applications such as electrical power

generation and marine power. About 600 of these engines were eventually produced

and saw successful service for more than two decades. Figure I-2 shows the SIGMA

GS-34. General Motors developed a similar machine - the GM-14, but with limited

success, and tested a much smaller machine – the GMR 4-4 ’Hyprex – for

automotive application, but this too was unsuccessful. Ford also experimented with

free-piston gasifiers without commercial success.


Figure I-2 Partial cut-away diagram of the SIGMA Type GS-34 Free-Piston

Gasifier [8]


I-2 Hydraulic free-piston engines

I-2-1 INNAS

In the early 1990’s, Dutch company Innas began developing a single ended direct

injected compression ignition hydraulic free-piston engine. The technical features

and measured performance of the INNAS free-piston engine (named Chiron) is

described by Achten in [7]. It has a nominal maximum power output of 17kW to a

hydraulic rail pressure of 260-320 bar. Output power control is achieved by pulse

pause modulation. Indicated efficiency is around 50% and net effective efficiency is

above 32% for the 25-100% load range. The Chiron engine as shown in Figure I-3 is

a mature prototype with demonstrably good piston control and performance.

However there has been little further published information on this engine since the

year 2000. NOAX continues to market and develop hydraulic hybrid vehicle

systems for INNAS. The INNAS website [65] provides some basic information.

INNAS and NOAX have been involved in the EU supported Free-piston Energy

Converter (FEPC) project since late 2002.

Figure I-3 INNAS “Chiron” hydraulic free-piston engine [7]

I-2-2 Tampere and Helsinki Universities of Technology

In 1999 Tampere University of Technology designed and built a prototype opposed

cylinder, hydraulic free-piston engine based on patents held by Sampower Ltd Oy.


This engine is shown in Figure I-4. Experimental results are reported by Tikkanen et

al. in [113]. Similar to the INNAS engine (above), this is a loop scavenged two-

stroke (using cylinder wall ports) with direct injection and compression ignition

however, it is an opposed cylinder machine. Measured output power was about

11kW with a net efficiency of about 20%.

Figure I-4 Tampere University of Technology hydraulic free-piston engine

prototype “Emma2” [113]

Combustion was found to be fast enough (about 2.5ms) and reliable. Some

asymmetry in the combustion pressures was evident, and this was attributed to un-

equal fuel injection, asymmetric pump design and manufacturing tolerances in the

cylinder assemblies.

Helsinki University of Technology undertook a simulation effort to analyse and

improve the engine [74].

A revised prototype (Figure I-5) was constructed with largely similar dimensions to

the initial prototype. The power was doubled by lightening the piston assembly

(increasing the operating frequency), slightly increasing the effective stroke, and

improvements to the hydraulic pump, scavenging and control system. Tikkanen and

Vilenius [114] report a control system designed for operating this machine. The

controller was tested on a Matlab/Simulink model of the engine and showed good

ability to control compression ratio, provided that the rate of change in load level

was kept within certain limits. No experimental results were reported.


Figure I-5 Third Generation prototype engine [114]

I-2-3 Toyohashi University of Technology

Hibi and Ito [63] report the latest experimental results from a long running hydraulic

free-piston engine project. The engine described is an opposed piston, direct

injected diesel compression ignition machine with a newly designed hydraulic circuit

and a peak power of almost 5kW. The combustion piston diameters are each 100mm

they each have an effective compression stroke of a little over 100mm. The piston

assemblies weigh 4.16kg each. The machine is intended to implement an engine-off

system utilising a hydraulic accumulator. The machine operates in single cycles,

allowing power output to be modulated, while maintaining optimum combustion

conditions. The maximum frequency reported was 7.25Hz (though judging by the

piston motion this could probably be increased significantly). The paper describes

the hydraulic circuit and control of the new engine in some detail, and gives

experimental hydraulic and gas pressures throughout an engine cycle. Triggering of

each engine cycle (from the piston’s holding positions at the outward position) is

activated when hydraulic pressure in the main accumulator falls below a certain

threshold.

Results showed that the net efficiency of the machine was substantially constant over

the full range of load level, staying at 31 1 percent, which though unexceptional,

is significant since the part load efficiency does not fall. A cross section of the

machine is shown in Figure I-6.


Figure I-6 Cutaway view of the Toyohashi university of Technology hydraulic

free-piston engine [63]

I-2-4 U.S. EPA, National Vehicle and Fuel Emissions Laboratory

The U.S. EPA engaged in an intensive free-piston engine development program as

part of their hydraulic hybrid vehicle program. Engine and powertrain development

company FEV were also contracted to this project and brought existing expertise

[57]. Brusstar et al. [32] describe the design and operating performance of two

prototype hydraulic free-piston engines,. The engines were both based on an

opposed cylinder module. One engine was a two cylinder two stroke version with a

rated output of 22kW, while the other was a six cylinder four stroke with a rated

output of 54kW. Both engines were compression ignition, utilising high pressure

direct injection or optional port fuel injection for HCCI operation. Scavenging in the

two-stroke was uni-flow via inlet cylinder ports and cam or hydraulic driven exhaust

valves. Valving in the four stroke version was hydraulics driven. Significantly, both

engines were operated for extended periods and the four stroke version demonstrated

an estimated net efficiency of 34-39% over much of the load range. Figure I-7

shows a sketch of the six cylinder four stroke, demonstrating reasonably flat

packaging.


Figure I-7 EPA six-cylinder four-stroke FPE [32]

I-2-5 Beijing Institute of Technology

Wu et al. describe the design and operation of a single ended diesel hydraulic free-

piston engine [126]. The hydraulic circuit is described in some detail and

preliminary experimental results are given for motored operation. Various design

issues are discussed such as the need for fast acting, high flow hydraulic valves, and

a control system capable of dealing with misfires. Contrary to the control method of

the EPA machine (above), which used controllable check valves to vary hydraulic

load at arbitrary operating pressures, this engine will vary the fuel quantity to

balance against the hydraulic load at different operating pressures. A numerical

model of the machine including piston dynamics, combustion cylinder

thermodynamics and hydraulic system dynamics is also described and simulation

results are presented.


I-3 Electric free-piston engines

I-3-1 University of West Virginia

West Virginia University have had a long running free-piston engine project that

attracted funding from the US army and has been the subject of several post-graduate

theses. Nandkumar [91] describes the early work on a spark ignition prototype build

around chainsaw engine pistons which had a bore of 36.5mm and stroke of about

40mm. Gasoline mixed with lubricating oil was injected near the cylinder ports.

The engine was initially tested with a rudimentary friction brake on the translator

assembly to extract mechanical work. It seems that fuelling and throttling levels

were set prior to individual test runs, and were not dynamically controlled during a

run. Due to the shape of the combustion chamber, the maximum achievable

compression ratio was about 8:1 since any further travel of the piston would result in

it striking the head. To prevent over-stroke ignition timing was radically advanced.

Somewhat surprisingly, the engine could run stably from zero load right up to the

stalling load, without any piston control, due to the self-regulating nature of the early

ignition, which automatically reduced thermodynamic efficiency when there was

little load on the translator. Presumably the problem of limited compression ratio

was only exacerbated under throttled operation. A permanent magnet alternator was

later used in place of the friction brake as a passive (uncontrolled) load.

Based on the experimental results and numerical modelling, a second generation

prototype was planned and built. Houdyschell [64] describes the initial design of the

engine which used significantly larger pistons (75mm diameter) and a total stroke of

about 70mm. Fuel was direct injection diesel or kerosene, though only kerosene was

used since the engine required a spark ignition stroke to start. The new engine was

designed to operate at higher frequency and compression ratio. It is shown in Figure

I-8. Due to regular catastrophic failure of various parts of the engine, Tóth-Nagy

[115] writes “The engine development was a continuous improving and redesigning

process”. Nevertheless, sufficient test results were obtained to do a fairly in-depth

analysis of the characteristics of the engine.


Figure I-8 West Virginia University, second generation linear engine prototype

(2-stroke) [115]

Again, no active load control was used on this engine, and it was initially run with

the same friction brake concept as the first prototype. It thus suffered like its

predecessor from over-stroke at low load, and peak cylinder pressures as high as

30Mpa were recorded. A closed loop fuel control was proposed to allow fuelling to

better match the applied load.

Petreanu [97] simulated a concept opposed four cylinder, four-stroke design, in an

attempt to circumvent the problems he considers inherent in the two stroke concept.

He states “The main problem of two-stroke cylinder processes is that they offer low

fuel efficiency due to the imperfect scavenging processes.” The proposed layout of

the four stroke engine is shown in Figure I-9.


Figure I-9 West Virginia University, four stroke concept [97]

Shoukry [105] reports more numerical modelling work on the two stroke concept,

finding among other things that shorter strokes lead to higher power operation but

reduced efficiency.

I-3-2 Sandia National Laboratories

The Sandia free-piston project grew out of research into hydrogen and high

compression ratio piston engines. In [121] van Blarigan reports the results of

combustion experiments using a rapid compression expansion machine (RCEM).

Fuels tested were propane, natural gas, hydrogen, methanol, npentane, hexane, n-

heptane, and isooctane. Ideal case indicated thermal efficiency of natural gas and

propane were the best at about 56% and did not increase significantly at higher

compression ratios. The good performance of these fuels is attributed to the very

fast (single step) burn characteristic giving effectively constant volume combustion.

The equivalence ratio used in the tests varied slightly depending on the fuel but all

were around 0.35 (lean). The main conclusions of the study are as follows:

Compression ratios as high as 30:1 are possible before the onset of auto-

ignition – though fuel, initial charge temperature and piston speed influence

the timing of combustion.

A number of fuels were capable of very rapid combustion – approximating

the constant volume “ideal”.


Over-compression due to premature combustion did not appear to adversely

affect cycle efficiency, but did tend to produce higher NOx emissions.

NOx could be controlled by lean operation or dilution with exhaust gas.

Different fuels had very different combustion rates. In order to ensure

complete combustion of some fuels, increased CR, increased initial charge

temp or slower piston motion could be employed.

Van Blarigan suggests that the main reason for the lack of expected efficiency

improvement at higher CR may be time loss (ie non constant volume combustion).

He dismisses increased heat loss as a primary cause since several tests with an

uncoated piston crown and cylinder head showed decreased efficiency of only 5%.

He did not however seem to consider the potential detrimental influence of increased

crevice volume flows at higher compression ratios.

Scott Goldsborough [58] reports an extensive CFD modelling project which he

conducted for his PhD thesis at the Colorado State University on the planned free-

piston engine shown in Figure I-10, a HCCI, port fuelled engine.

Figure I-10 First Sandia Free-piston Linear Alternator concept [116]

Goldsbrough’s 3D simulation gave best net cycle efficiency at 52%. While the

efficiency figures are useful for comparison between cases, Goldsborough

apparently did not account for blowby and crevice volume flow.

Soon after this during modelling work with the NASA [118] it was found that a

turbocharger would provide sufficient energy for pressurising the inlet air to drive

scavenging.

Soon after the completion of Goldsbrough’s modelling efforts on an opposed

cylinder, uni-flow layout (using inlet ports and exhaust valves) an opposed piston

layout was proposed having a single central combustion chamber as shown in Figure

I-11.


Figure I-11 Sandia opposed piston layout [119]

The chief benefits of the opposed piston layout are inherent mechanical balance

when the pistons are synchronised, and the opportunity to utilise a central

combustion chamber with inlet and exhaust ports at opposite ends. This gives uni-

flow scavenging without the need for any poppet valves. On the other hand,

accurate piston synchronisation is essential, the overall length of the engine is

increased over a single piston concept, two extra bounce cylinders are required, as

are two generators. It is not clear how much of the design of this prototype is

intended to embody a final commercial engine layout. Features of this engine design

are:

twin linear alternators connected electrically in parallel to effect the piston

synchronisation. The load on the alternator coils is resistive at present (not

actively switched), and is constant at about 15kW in total. This is

considered necessary to maintain the synchronisation.

High pressure air injection into bounce chambers to achieve piston motion

during un-powered and low powered cycles.

Port fuel injection (low delivery ratio essential to avoid fuel short circuiting)

The compression ratio control is intriguing. Under “motoring” conditions, over

15kW of compressed air supply is required to drive the piston oscillation against the

constant generator load (the linear generator is passive). If fuel injection is

increased, increasing combustion energy allows the vent pressure in the bounce

chamber to be increased, presumably to the point where no extra air injection is

required. One potential criticism of the concept is losses incurred in the bounce

chambers form heat transfer, gas leakage, valve flow. Careful design should be able

to minimise these.


Figure I-12 Sandia bounce chamber detail showing compressed air injection

valves on the left and vent ports on the right [119]

The aims of the project are [119]

To study the effects of continuous operation (i.e. gas exchange) on indicated

thermal efficiency and emissions at high compression ratios (~20-40:1)

Concept validation of passively coupling the opposed free-pistons via the

linear alternators connected to a common load to maintain piston

synchronization

Proof of principle of electronic variable compression ratio control

Figure I-13 shows the complete opposed piston prototype, awaiting final installation

of the bounce chamber injectors and latch mechanisms. It will provide a platform

for testing the free-piston operating envelope over varying levels of boost,

equivalence ratio, and alternative fuels.


Figure I-13 Sandia Opposed Piston Free-piston Engine [120]

Current project partners are Los Alamos National Laboratory (engine and generator

modelling), General Motors/University of Michigan Collaborative Research

Laboratory (engine modelling) and Stanford University.

Figure I-14 gives a timeline of the project. The project has also included a

substantial effort in linear generator design.

Figure I-14 Timeline of Sandia free-piston engine project [119]

I-3-3 FPEC

A European project consortium with funding from the Swedish Energy Agency and

the European Commission has been developing a free-piston engine since 2002

called the “Free-piston Energy Converter” (FPEC). A completed prototype is shown

in Figure I-15. According to [70] the engine has a bore of 102mm and a nominal


stroke of about 127mm, and will operate at a high compression ratio of 26:1 with

50% EGR. [78] provides a good summary of the results of the project as of 2005,

just prior to the commencement of experimental testing. The engine is opposed

cylinder, diesel direct injected (to operate with early injection for HCCI-like

combustion). Chalmers University of Technology, Sweden have conducted

extensive modelling of the combustion and combustion related aspects – see for

example [51]. A tubular permanent magnet machine was designed by Sheffield

University and constructed by ABB Corporate Research, Sweden. [122] The electric

machine was predicted to be capable of operating at about 30kW continuously with

an efficiency of 93%. In parallel, the Royal Institute of Technology Stockholm,

Sweden (KTH) have designed and built an alternative electric machine [40] using an

unusual transverse flux design. French research organisation IFP conducted initial

modelling work to verify and the expected performance of the scavenging and

determine optimum valve timing and port layout [70]. Dutch companies INNAS and

NOAX with hydraulic free-piston engine experience are project members. Volvo

Technology Corporation, Sweden is also a project partner.

Figure I-15 FPEC prototype electric generator – opposed cylinder, two-stroke

diesel with pnumatic exhaust valves [78]

I-3-4 University of Newcastle upon the Tyne

The University of Newcastle upon Tyne began a free-piston engine project in 1999

[93]. In the last couple of years, Mikalsen and Roskilly have published numerous

papers detailing the modelling of a proposed two stroke, single ended electric free-

piston engine [80].


I-3-5 Pempek

Australian industrial electronics company Pempek Systems Pty. Ltd. developed and

tested a compact two stroke, opposed cylinder engine[96]. The details of this project

are described in Chapter 2.

I-3-6 Korea Institute of Energy

An opposed cylinder spark ignition engine prototype built by the Korea Institute of

Energy is described in [125]. The engine was tested with both CNG and Hydrogen

fuel. Variability in the stroke caused some instability in operation. The control

method is not disclosed, though it seems that the linear generator is a passive

machine, and fuel mass is the control parameter. A revised design is proposed which

replaces the original ported loop scavenging with a uni-flow system in which poppet

valves in the cylinder head evacuate the exhaust gases and the generator is modified

to provide integral supercharging of the inlet air.

Figure I-16 Korea Institute of Energy opposed cylinder engine [125]

I-3-7 Malaysian Ministry of Science, Technology and Environment

A free-piston engine project funded by the Ministry of Science, Technology and

Environment, Malaysia is reported in [17]. This engine is a single ended spark

ignition machine with air cushion bounce chamber and was built using the piston and

cylinder head of a two stroke motorcycle engine in similar fashion to the University

of West Virginia machines (described above). The engine was operated with no load

and the piston motion analysed using a high speed camera. A related project [89]

describes a numerical model of a similar engine utilising a linear electric machine for

power extraction. Work on an appropriate linear electric machine for the free-piston

engine is reported from the University of Malaya [61] and the Universiti Teknologi

PETRONAS, Malaysia [133]


I-3-8 Loughborough/Sheffield/Lotus

Loughborough University partnered with Sheffield University and Lotus

Engineering Ltd. in 2005 under a UK government EPSRC grant to develop a four

stroke free-piston engine [45].

I-3-9 Shanghai Jiao Tong University

Li et al. [75] report a numerical investigation of an opposed cylinder machine with

linear alternator with special emphasis on HCCI operation. A chemical kinetics

model (Chemkin/Senkin) was used for the in-cylinder combustion process which

was assumed homogenous, adiabatic and perfectly sealed. The force characteristic

of the linear alternator was analysed using the Finite Element Method, and this

compared favourably with simple analytical expressions which were incorporated in

the engine model. The alternator was modelled with an uncontrolled resistive load.

The piston dynamics model was implemented with Matlab/Simulink. The study

traced the relationships between generator load, compression ratio and equivalence

ratio. The shorter residence time at TDC compared to a cranked engine was noted,

with possible benefits being reduced NOx formation and reduced heat transfer. It is

unclear if any closed loop control is implemented in the model (such as equivalence

ratio control), though given the inherent damping characteristic of a resistive load,

closed loop control may not be necessary if the combustion strength is fairly

constant.

I-3-10 Nanjing University of Science and Technology

The Nanjing University of Science and Technology [128] report the design and

modelling of a novel four-stroke, single ended, spark ignition free-piston engine as

sketched in Figure I-17. Springs attached to the mover provide the energy storage

for exhaust the intake strokes, while the large energy requirement of the compression

stroke is provided by the linear electric machine during this stroke. The control

system uses electromagnetic force as the main control variable, and treats fuel

delivery as an essentially fixed input. Needless to say, the electric machine must

allow bidirectional energy transfer. Simulation results of piston dynamics show the

viability of this approach.


Figure I-17 Four-stroke free-piston engine concept by Nanjing University of

Science and Technology [128]

I-3-11 German Aerospace Center

A project at the German Aerospace Center report preliminary modelling and testing

of various components of a free-piston linear generator [60]. The proposed engine is

a single ended machine with electromagnetic inlet and exhaust valves. Notably, a

permanent magnet linear generator has been constructed and tested, showing good

performance due to effective cooling. Efficiency of the engine’s gas spring has been

measured at 90-95% depending on engine speed. Heat loss to walls was found to be

the primary loss mode with blowby accounting for only 10% of loss. Piston motion

control is based on generator load control. The prototype engine is shown in Figure

I-18.

Figure I-18 German Aerospace Center free-piston prototype on test bench [60]


I-4 Summary of linear generator projects

A suitable linear generator is one of the most challenging design problems facing the

development of electric free-piston engines. High efficiency and high power-to-

weight ratio must be combined in the one machine, which must also be able to

withstand the mechanical shock of repeated high mover accelerations. Many of the

electric free-piston engine projects cited above are devoting significant effort in this

area. While a detailed survey of this field is beyond the scope of this thesis, a brief

(and no doubt, incomplete) summary of some linear generator projects is given

below:

1999 As part of the West Virginia University (WVU) Project, [34, 48] describe the

design and testing of a permanent magnet linear alternator. Maximum power output

at 23Hz was 316W. The alternator was tested with the opposed cylinder engine and

was connected to an adjustable, purely resistive load. A cross section of the

alternator is shown in Figure I-19. Rerkpreedapong [100] describes the design and

modelling of an alternative machine using a moving iron topology. This has the

advantage of placing the permanent magnets in the stator, thus allowing improved

cooling and protection from the high acceleration loads of the mover. However the

mass of the mover is increased compared to moving magnet designs, so power

density is lower due to reduced mover frequency. Predicted efficiency was 79%.

Figure I-19 Linear alternator built at WVU [34]

2006 Faiz et al. [47] from the University of Tehran describe the modelling and

testing of a self-excited reciprocating generator with flux concentrators. The merits

of the design are a lightweight mover, low maintenance, and robust structure.

Conducting plates are inserted into the stator slots to achieve flux concentration by

eddy currents. The presence of the plates increases the performance of the engine


significantly; however the efficiency is not as good as permanent magnet type

generators.

The Sandia free-piston engine project designed and built two similar linear

alternators. One was developed in-house (Figure I-20). The stator was made from

1600 ground tapered laminations, and each of the 25 coils contained 78 turns of

square copper wire. Nominal power output is 40kW at 94% efficiency. A second

alternator was made by Magnaquench. The stator of this machine was made from

powdered iron in an adhesive matrix (Figure I-21). The actual performance of these

machines is unknown, however it appears that the magnaquench machine is being

used in the latest dual alternator machine described in [119].

Figure I-20 Sandia linear alternator design [117]

Figure I-21 Magnaquench linear alternator stator [117]

Work at the University of Sheffield on a machine for the Free-piston Energy

Converter (FPEC) has culminated in the machine decribed in [122]. The machine

has three phases and employs a quasi-Halbach permanent magnet arrangement on

the mover and since no back iron is required, the permanent magnets are mounted


directly on a non-magnetic support tube. The machine has a high power density, low

cogging torque and has a simplified manufacturing process. The stator is made in

modules with flat laminations.

Figure I-22 University of Sheffield FPEC linear permanent magnet generator

[122]

An alternative transverse flux machine has been developed at the Royal Institute of

Technology, Sweden [16, 39, 40] as shown in Figure I-23. Simulation work on a

more conventional longitudinal flux machine is also reported [131].

Figure I-23 Royal Institute of Technology Stockholm, Sweden transverse flux

permanent magnet generator [40]


[133] mentions a conventional surface magnet generator developed by Universiti

Teknologi PETRONAS (UTP) in collaboration with University Malaya (UM) and

University Kebangsaan Malaysia (UKM). [61] describes an alternative machine

from the UM using a quasi Halback permanent magnet arrangement to eliminate the

need for back iron.

The German Aerospace Center have constructed a prototype linear generator which

is mentioned briefly in [60].

Figure I-24 German Aerospace Center linear generator on the test stand [60]

I-5 Other free-piston engines

I-5-1 University of Minnesota

A numerical and experimental program at the University of Minnesota investigated

the viability of miniature free-piston engine with a power output of 10W. The

rationale behind the investigation was the superior energy density of liquid

hydrocarbon fuels compared with battery technology. Among the technical issues

present with miniature combustion engines, the work here focused on micro-

combustion [10], in particular HCCI combustion [11]. The characteristics of HCCI

combustion make it particularly advantageous for miniature engines. Since the

charge is ignited by compression heating, no ignition system is required. The

presence of multiple ignition points and well mixed charge ensure rapid, even

combustion, minimising combustion chamber wall quenching effects and allowing

high engine speeds. The use of the free-piston concept allows variable compression

ratio to be used as a combustion phasing control. Single shot experiments [12]

confirmed that a large alkane (heptane) at a lean equivalence ratio (0.25) could be

reliably combusted in a space 3mm in diameter and 0.3mm in length, with sufficient


speed to produce virtually constant volume combustion. Subsequent analysis of the

experimental data using a numerical model allowed piston blowby to be quantified.

Further details of the project (which ended in 2002) can be found at the project

website [9].

More recently, the centre for Compact and Efficient Fluid Power (CCEFP) have

reported design, modelling and preliminary experimental work on a miniature HCCI

air compressor free-piston engine [112]. The concept design is shown in Figure I-

25.

Figure I-25 Design concept for a miniature HCCI free-piston engine [112]

I-5-2 Kvaerner ASA and Norwegian University of Science and

Technology

Kvaerner ASA built a free-piston gasifier engine with exhaust gas turbine for power

extraction in a development effort aimed at marine applications as an alternative to

both gas turbines and traditional diesel engines. It consists of multiple single piston

units connected to common intake and exhaust manifolds. The engine takes some

design cues from the commercially successful Pescara/SIGMA free-piston gasifiers.

The Kvaerner prototype is a single piston unit with a nominal power output of 1MW.

A photograph of the engine rig is shown in Figure I-26. Control focuses on BDC

and TDC position control using fuel mass and bounce chamber air mass respectively

as control parameters. Frequency can be varied slightly during operation by

changing the stroke length, providing a mechanism of keeping multiple cylinders

properly synchronised. Johansen et al. [66] describe the salient details of the engine,


test rig and control system, and give some experimental results which demonstrate

that the control system performs well in maintaining proper piston motion control

over the load range. In a companion paper, Johansen et al. [67] describe a

mathematical model of the engine dynamics which is used to derive an engine

control system structure.

Figure I-26 Photo of Kvaerner test cylinder unit

I-5-3 Marquette University

Bosman and Goldsbourough build on Goldsbourough’s previous work with Sandia

to do preliminary scavenging and combustion modelling of a simple two-stroke free-

piston engine with opposed cylinders designed for driving hydraulic or pneumatic

loads [31]. The engine uses a passive inlet valve in the cylinder head and exhaust

ports at the bottom section of the cylinder. Small engines are envisaged; however

the design could be scaled up for automotive applications. Initial scavenging

modelling reveals high residual exhaust fraction due to entrainment to the lee of the

inlet valve, and non-optimal valve and port timing.


I-5-4 Vanderbilt University

A free-piston air compressor for applications of human scale robotics has been

reported from the Laboratory for the Design and Control of Energetic Systems at

Vanderbilt University. A slug of fluid contained in a cylinder and on each end by a

flexible membrane forms a “liquid piston” with perfect sealing and negligible

friction. A combustion chamber on one end drives the liquid piston to pump air into

a pressurised reservoir on the other end [102, 129]. Subsequent experimental and

modelling work revealed the need to slow down the dynamics of the system, so

rather than increase the piston mass, a high inertance piston was devised [123].

Figure I-27 Liquid piston air compressor prototype (high inertance model)

Vanderbilt University [73]

235

Appendix II

Specific heats of a reacting mixture

Specific heat is the amount of heat per unit mass required to raise the temperature of

a substance by one degree. Two distinct specific heats are defined, one at constant

pressure and the other at constant volume.

However, for reacting mixtures (in chemical equilibrium), there is another important

intensive property to consider, and this is the relative species fractions.

Traditionally, specific heat for a reacting mixture is defined for the case of shifting

(chemical) equilibrium – that is, the specific heat that would be measured if an

experiment was conducted on the mixture in question. The derivation is as follows.

The internal energy of a mixture of species is given by

Differentiating WRT to temperature gives

Likewise, for a mixture of ideal gases, the enthalpy of a mixture is given by

Differentiating WRT to temperature gives

Thus for a reacting mixture the true specific heats are defined as

Appendix II Specific heats of a reacting mixture 236

Then the ratio of specific heats is

and

It is useful however, to make an alternative definition for specific heat, where the

chemical composition of the fluid is held constant (frozen). In this case, the

chemical reaction term is zero, and the frozen specific heats become.

The ratio of frozen specific heats is

Then

Frozen specific heats are used throughout this thesis for reacting mixtures. This is

because of the choice to model reacting mixtures explicitly, with each species

contributing to the energy balance of a mixture. All chemical reaction effects (be

they dissociation in shifting equilibrium or fuel oxidisation etc.) are written into the

energy equation explicitly, rather than being subsumed into the specific heat.

237

Appendix III

Further applications of the energy equation

The form of the energy equation (3-2) used in this engine model was chosen for its

ability to conveniently and accurately model the thermodynamics of arbitrary

reacting mixtures. The following section offers alternative forms of the equation

which may be useful for certain applications. Most significantly, it is shown how the

energy equation can be used “in reverse” to analyse cylinder pressure data for

combustion, heat transfer and blowby characteristics.

The energy equation for an open control volume containing several chemical species

is given by equation (3-2).

Expanding the flow enthalpy term the energy equation becomes.

Using and separating the species mass flow out of the term

and relocating it to the flow enthalpy term yields

(III-1)

where the now represents the change in energy due to chemical reactions

only. If the specific heat of each species is reasonably constant over the

temperature range between the bulk volume temperature and the flow temperatures

then internal energy terms of the mass flows can be substituted away.


(III-2)

Evaluating heat release

Equation (III-2) can be written in terms of pressure so as to estimate the chemical

reaction heat release from experimental engine data.

The ideal gas equation is:

Differentiating with respect to time gives

Given that

and

Then

And differentiating WRT time

So the ideal gas equation in rate form becomes

(III-3)


So assuming the cylinder can be modelled as an ideal gas, equation (III-3) is used to

substitute for in Equation (III-2) and bringing the chemical energy change to

the LHS gives

Writing in terms of the frozen specific heat ratio and neglecting the second flow

enthalpy term gives.

Note that for an exothermic reaction, the chemical energy change is negative. Note

also mass flow and heat transfer is positive into the cylinder. P and can be

obtained from pressure transducer data. Likewise the piston motion must be known

to find V and . The chemical heat release rate can be numerically integrated

to determine the cumulative total heat release. If mass flows (eg fuel injection,

crevice flows, blowby, valve leakage) and heat transfer are neglected then the

resulting heat release is termed apparent.

In heat release calculations the term is often neglected. Thus the third term

on the RHS of equation (III-4) is not usually considered. Figure III- shows that for a

mixture in shifting chemical equilibrium significant variation in mixture R does not

happen until the mixture is heated above about 2500K. Also, the gas constant of the

un-reacted mixture (279 J/kg/K in this case) or pure air (287 J/kg/K) is not too

dissimilar to that of the burned state.

(III-4)


Figure III-1 Variation of R with temperature for products of combustion in

chemical equilibrium

280

300

320

340

360

380

400

0 1000 2000 3000 4000 5000

1 bar 10 bar

100 bar

0T (K)

R (J

/kg/

K)

Mixture made from 75 parts N2, 15 parts O2 and 5 parts n-octane (C8H18) by mass

241

Appendix IV

Tables of Thermodynamic Properties

Table IV-1 Enthalpy of some combustion products Enthalpy Reference Temp=298.15 Absolute enthalpy (J/mol) Standard state Pressure = 0.1Mpa

Source JANAF Thermochemical Tables (3rd Edition) T (K) CO CO2 H H2 H2O N N2 NO O O2 OH M 28.0104 44.0098 1.00794 2.01588 18.01528 14.0067 28.0134 30.0061 15.9994 31.9988 17.00734

100 -116296 -399978 213880 -5468 -248441 468564 -5768 84218 244655 -5779 32848 200 -113385 -396936 215959 -2774 -245108 470643 -2857 87340 246987 -2868 36011 300 -110473 -393453 218037 53 -241764 472721 54 90346 249214 54 39042 400 -107551 -389519 220116 2959 -238374 474800 2971 93331 251380 3025 42022 500 -104596 -385217 222195 5882 -234901 476879 5911 96350 253516 6084 44979 600 -101585 -380615 224273 8811 -231325 478957 8894 99435 255635 9244 47930 700 -98504 -375768 226352 11749 -227634 481036 11937 102598 257743 12499 50889 800 -95350 -370716 228430 14702 -223824 483114 15046 105839 259844 15835 53867 900 -92126 -365492 230509 17676 -219888 485193 18223 109149 261940 19241 56875

1000 -88837 -360125 232588 20680 -215826 487272 21463 112520 264033 22703 59922 1100 -85492 -354638 234666 23719 -211635 489350 24760 115944 266123 26212 63011 1200 -82097 -349049 236745 26797 -207320 491429 28109 119411 268212 29761 66147 1300 -78659 -343374 238823 29918 -202884 493507 31503 122917 270299 33344 69329 1400 -75184 -337626 240902 33082 -198333 495586 34936 126455 272385 36957 72556 1500 -71677 -331817 242981 36290 -193675 497665 38405 130020 274469 40599 75826 1600 -68142 -325953 245059 39541 -188918 499743 41904 133610 276554 44266 79138 1700 -64582 -320042 247138 42835 -184068 501822 45429 137220 278637 47958 82489 1800 -61001 -314091 249216 46169 -179133 503901 48978 140848 280720 51673 85876 1900 -57401 -308103 251295 49541 -174120 505979 52548 144492 282803 55413 89297 2000 -53783 -302083 253374 52951 -169036 508058 56137 148150 284886 59175 92749 2100 -50151 -296034 255452 56397 -163885 510137 59742 151821 286969 62961 96230 2200 -46506 -289960 257531 59876 -158673 512217 63361 155503 289051 66769 99739 2300 -42848 -283862 259609 63387 -153405 514297 66995 159195 291135 70600 103272 2400 -39179 -277743 261688 66928 -148085 516378 70640 162897 293218 74453 106828 2500 -35500 -271605 263767 70498 -142718 518460 74296 166607 295303 78328 110406 2600 -31812 -265449 265845 74096 -137306 520543 77963 170325 297389 82224 114004 2700 -28116 -259276 267924 77720 -131853 522628 81639 174050 299476 86141 117620 2800 -24411 -253089 270003 81369 -126362 524716 85323 177782 301564 90079 121254 2900 -20700 -246886 272081 85043 -120836 526807 89015 181520 303654 94036 124905 3000 -16981 -240670 274160 88740 -115277 528901 92715 185264 305747 98013 128571 3100 -13256 -234441 276238 92460 -109687 531000 96421 189013 307842 102009 132252 3200 -9526 -228201 278317 96202 -104069 533103 100134 192768 309940 106023 135947 3300 -5789 -221949 280396 99966 -98423 535213 103852 196527 312040 110054 139654 3400 -2047 -215686 282474 103750 -92753 537329 107577 200291 314144 114102 143374 3500 1700 -209413 284553 107555 -87058 539452 111306 204059 316252 118165 147106 3600 5451 -203129 286631 111380 -81341 541585 115041 207832 318363 122245 150850 3700 9208 -196836 288710 115224 -75604 543726 118781 211609 320478 126339 154604 3800 12969 -190533 290789 119089 -69846 545877 122525 215389 322597 130447 158368 3900 16734 -184221 292867 122972 -64069 548040 126274 219174 324720 134569 162142 4000 20504 -177900 294946 126874 -58274 550215 130027 222962 326848 138705 165926 4100 24277 -171571 297024 130795 -52463 552402 133784 226753 328981 142854 169719 4200 28055 -165232 299103 134734 -46635 554603 137545 230548 331118 147015 173521 4300 31836 -158885 301182 138692 -40792 556819 141309 234347 333260 151188 177332 4400 35621 -152531 303260 142667 -34934 559050 145078 238148 335407 155374 181151 4500 39410 -146168 305339 146660 -29062 561297 148850 241953 337559 159572 184978 4600 43202 -139797 307417 150670 -23176 563560 152625 245760 339716 163783 188814 4700 46998 -133419 309496 154698 -17278 565841 156405 249571 341878 168005 192657 4800 50797 -127033 311575 158741 -11368 568140 160187 253385 344045 172240 196508 4900 54599 -120640 313653 162801 -5446 570458 163973 257201 346218 176488 200367 5000 58405 -114239 315732 166876 487 572794 167763 261021 348395 180749 204233 5100 62214 -107831 317810 170967 6432 575150 171556 264843 350578 185023 208107 5200 66026 -101413 319889 175071 12389 577526 175352 268668 352765 189311 211988 5300 69841 -94987 321968 179190 18358 579921 179152 272495 354957 193614 215876 5400 73660 -88551 324046 183322 24338 582338 182955 276325 357155 197933 219771 5500 77481 -82106 326125 187465 30331 584775 186761 280158 359357 202267 223674 5600 81306 -75652 328203 191621 36335 587233 190571 283994 361564 206618 227584 5700 85134 -69188 330282 195787 42351 589711 194384 287832 363775 210987 231501 5800 88965 -62716 332361 199963 48378 592211 198201 291672 365991 215375 235425 5900 92799 -56234 334439 204148 54418 594732 202023 295515 368212 219782 239356 6000 96636 -49743 336518 208341 60469 597273 205848 299361 370437 224210 243295


Table IV-2 Specific heat of some combustion products Enthalpy Reference Temp=298.15 Specific heat at constant pressure (J/mol/K) Standard state Pres. = 0.1Mpa

Source JANAF Thermochemical Tables (3rd Edition) T (K) CO CO2 H H2 H2O N N2 NO O O2 OH M 28.0104 44.0098 1.00794 2.01588 18.01528 14.0067 28.0134 30.0061 15.9994 31.9988 17.00734

100 29.104 29.208 20.786 28.154 33.299 20.786 29.104 32.302 23.703 29.106 32.627 200 29.108 32.359 20.786 27.447 33.349 20.786 29.107 30.420 22.734 29.126 30.777 300 29.142 37.221 20.786 28.849 33.596 20.786 29.125 29.841 21.901 29.385 29.977 400 29.342 41.325 20.786 29.181 34.262 20.786 29.249 29.944 21.482 30.106 29.650 500 29.794 44.627 20.786 29.260 35.226 20.786 29.580 30.486 21.257 31.091 29.521 600 30.443 47.321 20.786 29.327 36.325 20.786 30.110 31.238 21.124 32.090 29.527 700 31.171 49.564 20.786 29.441 37.495 20.786 30.754 32.028 21.040 32.981 29.663 800 31.899 51.434 20.786 29.624 38.721 20.786 31.433 32.767 20.984 33.733 29.917 900 32.577 52.999 20.786 29.881 39.987 20.786 32.090 33.422 20.944 34.355 30.264

1000 33.183 54.308 20.786 30.205 41.268 20.786 32.697 33.987 20.915 34.870 30.676 1100 33.710 55.409 20.786 30.581 42.536 20.786 33.241 34.468 20.893 35.300 31.124 1200 34.175 56.342 20.786 30.992 43.768 20.786 33.723 34.877 20.877 35.667 31.586 1300 34.572 57.137 20.786 31.423 44.945 20.786 34.147 35.226 20.864 35.988 32.046 1400 34.920 57.802 20.786 31.861 46.054 20.786 34.518 35.524 20.853 36.277 32.492 1500 35.217 58.379 20.786 32.298 47.090 20.786 34.843 35.780 20.845 36.544 32.917 1600 35.480 58.886 20.786 32.725 48.050 20.786 35.128 36.002 20.838 36.796 33.319 1700 35.710 59.317 20.786 33.139 48.935 20.786 35.378 36.195 20.833 37.040 33.694 1800 35.911 59.701 20.786 33.537 49.749 20.787 35.600 36.364 20.830 37.277 34.044 1900 36.091 60.049 20.786 33.917 50.496 20.788 35.796 36.514 20.827 37.510 34.369 2000 36.250 60.350 20.786 34.280 51.180 20.790 35.971 36.647 20.826 37.741 34.670 2100 36.392 60.622 20.786 34.624 51.823 20.793 36.126 36.767 20.827 37.969 34.950 2200 36.518 60.865 20.786 34.952 52.408 20.797 36.268 36.874 20.830 38.195 35.209 2300 36.635 61.086 20.786 35.263 52.947 20.804 36.395 36.971 20.835 38.419 35.449 2400 36.740 61.287 20.786 35.559 53.444 20.813 36.511 37.060 20.841 38.639 35.673 2500 36.836 61.471 20.786 35.842 53.904 20.826 36.616 37.141 20.851 38.856 35.881 2600 36.924 61.647 20.786 36.111 54.329 20.843 36.713 37.216 20.862 39.068 36.075 2700 37.003 61.802 20.786 36.370 54.723 20.864 36.801 37.285 20.877 39.276 36.256 2800 37.083 61.952 20.786 36.618 55.089 20.891 36.883 37.350 20.894 39.478 36.426 2900 37.150 62.095 20.786 36.856 55.430 20.924 36.959 37.410 20.914 39.674 36.586 3000 37.217 62.229 20.786 37.087 55.748 20.963 37.030 37.466 20.937 39.864 36.736 3100 37.279 62.347 20.786 37.311 56.044 21.010 37.096 37.519 20.963 40.048 36.878 3200 37.338 62.462 20.786 37.528 56.323 21.064 37.158 37.570 20.991 40.225 37.013 3300 37.392 62.573 20.786 37.740 56.583 21.126 37.216 37.617 21.022 40.395 37.140 3400 37.443 62.681 20.786 37.946 56.828 21.197 37.271 37.663 21.056 40.559 37.261 3500 37.493 62.785 20.786 38.149 57.058 21.277 37.323 37.706 21.092 40.716 37.376 3600 37.543 62.884 20.786 38.348 57.276 21.365 37.373 37.747 21.130 40.868 37.486 3700 37.589 62.980 20.786 38.544 57.480 21.463 37.420 37.787 21.170 41.013 37.592 3800 37.631 63.074 20.786 38.738 57.675 21.569 37.465 37.825 21.213 41.154 37.693 3900 37.673 63.166 20.786 38.928 57.859 21.685 37.508 37.862 21.257 41.289 37.791 4000 37.715 63.254 20.786 39.116 58.033 21.809 37.550 37.898 21.302 41.421 37.885 4100 37.756 63.341 20.786 39.301 58.199 21.941 37.590 37.933 21.349 41.549 37.976 4200 37.794 63.426 20.786 39.484 58.357 22.082 37.629 37.966 21.397 41.674 38.064 4300 37.832 63.509 20.786 39.665 58.507 22.231 37.666 37.999 21.445 41.798 38.150 4400 37.869 63.588 20.786 39.842 58.650 22.388 37.702 38.031 21.495 41.920 38.233 4500 37.903 63.667 20.786 40.017 58.787 22.551 37.738 38.062 21.545 42.042 38.315 4600 37.941 63.745 20.786 40.188 58.918 22.722 37.773 38.092 21.596 42.164 38.394 4700 37.974 63.823 20.786 40.355 59.044 22.899 37.808 38.122 21.647 42.287 38.472 4800 38.007 63.893 20.786 40.518 59.164 23.081 37.843 38.151 21.697 42.413 38.549 4900 38.041 63.968 20.786 40.676 59.275 23.269 37.878 38.180 21.748 42.542 38.625 5000 38.074 64.046 20.786 40.829 59.390 23.461 37.912 38.208 21.799 42.675 38.699 5100 38.104 64.128 20.786 40.976 59.509 23.658 37.947 38.235 21.849 42.813 38.773 5200 38.137 64.220 20.786 41.117 59.628 23.858 37.981 38.262 21.899 42.956 38.846 5300 38.171 64.312 20.786 41.252 59.746 24.061 38.013 38.289 21.949 43.105 38.919 5400 38.200 64.404 20.786 41.379 59.864 24.266 38.046 38.316 21.997 43.262 38.991 5500 38.233 64.496 20.786 41.498 59.982 24.474 38.080 38.342 22.045 43.426 39.062 5600 38.263 64.588 20.786 41.609 60.100 24.682 38.116 38.367 22.093 43.599 39.134 5700 38.296 64.680 20.786 41.712 60.218 24.892 38.154 38.393 22.139 43.781 39.206 5800 38.325 64.772 20.786 41.806 60.335 25.102 38.193 38.418 22.184 43.973 39.278 5900 38.355 64.865 20.786 41.890 60.453 25.312 38.234 38.443 22.229 44.175 39.350 6000 38.388 64.957 20.786 41.965 60.571 25.521 38.276 38.468 22.273 44.387 39.423


Table IV-1 and Table IV-2 list the absolute enthalpy and specific heats of 11

combustion species. The source is the JANAF thermochemical tables [38].

The absolute enthalpy is enthalpy relative to stable elements at the reference state

(0.1MPa, 298.15K). It was calculated as.

where is sensible enthalpy, is the reference temperature 298.15K, and is the

enthalpy of formation at the reference state.

Table IV-3 Properties of some fuels (various sources [30, 62, 107]) nOctane Methane Ethane Propane Methanol Ethanol Gasoline Gasoline2 Diesel

Formula C8H18 C1H4 C2H6 C3H8 C1H4O1 C2H6O1 C8.26H15.5 C7.76H13.1 C10.8H18.7 M(g/mol) 114.2222 16.0416 30.0674 44.0932 32.0406 46.0664 114.825 106.401 148.556

(J/mol) -208826 -74933 -84801 -104062 -201342 -235055 -200000 -200000 -350000 BP (K) 399.15 112.15 184.15 231.15 338.15 351.15 387.15 387.15 493.15 (J/mol) 34381 8181 14703 19048 35213 38512 43634 40432 39900 A1 38.03599 6.946013 3.908762 10.49418 13.07256 9.627878 -8.3736 12.5604 -0.83736 A2 0.592813 0.101864 0.183411 0.246922 0.119191 0.225725 0.632207 0.519163 0.879228 A3 -0.00023 -4.17E-05 -7.40E-05 -9.70E-05 -5.13E-05 -1.13E-04 -2.51E-04 -1.95E-04 -4.06E-04 A4 3.43E-08 6.42E-09 1.02E-08 1.46E-08 7.69E-09 1.89E-08 3.35E-08 2.47E-08 6.28E-08 B1 40.7229 64.5193 89.7211 109.3511 81.0948 112.3099 81.5531 81.5531 165.9032 B2 0.2825 0 0 0 0 0 0.5657 0.5657 0.6933

Table IV-3 lists properties of various fuels. The specific heat , in gas phase was

derived from various references [30, 62, 107] as polynomial functions of

temperature. Some massaging of the data was required to convert disparate units to

J/mol and to guess the values likely at high temperatures since the functions given

were typically only valid between 250 and 1200K. Likewise the liquid phase

specific heat , enthalpy of evaporation , boiling point and Lower Heating

Value (LHV) were obtained where available from [30, 107] and otherwise estimated

based on similar fuels. The molar mass M was calculated based on atomic formula.

The enthalpy of formation was calculated based on the LHV. The values used

for nine typical fuels are shown in Table IV-3. Other fuels can be added to the

database in the future if necessary.

Specific heats in gas and liquid phase are represented respectively by polynomial

functions as:

For gas, these are valid up to about 3000K.


Enthalpy of formation at standard state (298.15K, 0.1MPa) was calculated as

where LHV is in units of J/mol and is based on gas phase reactants and products.

J/mol

J/mol

The average composition of the fuel is

The absolute enthalpy of the fuel may be calculated by

Table IV-4 lists eight reactions and the formula for calculation the corresponding

equilibrium constant.

Table IV-4 Equilibrium equations

Reaction Equilibrium constant

1

2

3

4

5

6

7

8

The thermodynamic property is defined as

where is the molar specific entropy at the reference pressure (0.1Mpa) and is

the absolute enthalpy. Table IV-5 lists the equilibrium constant calculated for each

reaction using data from the JANAF tables [38].


Table IV-5 Equilibrium constants for selected reactions Enthalpy Reference Temp=298.15 Equilibrium Constant ln(K) Standard state Pres. = 0.1Mpa

Source data JANAF Thermochemical Tables (3rd Edition) Reactions as listed in Table IV-4

T (K) 1 2 3 4 5 6 7 8 100 -284.545 -328.876 44.3309 -89.5166 -583.597 -1121.84 -511.035 -213.950 200 -139.976 -159.696 19.7200 -43.0614 -285.452 -554.538 -250.170 -105.599 300 -91.6072 -103.059 11.4520 -27.4608 -185.722 -365.173 -162.917 -69.4185 400 -67.3224 -74.6700 7.34761 -19.6410 -135.708 -270.364 -119.159 -51.3152 500 -52.6913 -57.6163 4.92503 -14.9476 -105.622 -213.403 -92.8293 -40.4494 600 -42.8986 -46.2436 3.34505 -11.8238 -85.5184 -175.381 -75.2257 -33.2041 700 -35.8776 -38.1227 2.24509 -9.59954 -71.1288 -148.189 -62.6170 -28.0280 800 -30.5937 -32.0358 1.44206 -7.93788 -60.3171 -127.772 -53.1340 -24.1458 900 -26.4709 -27.3060 0.83505 -6.65105 -51.8939 -111.875 -45.7384 -21.1251

1000 -23.1632 -23.5266 0.36334 -5.62670 -45.1455 -99.1455 -39.8065 -18.7083 1100 -20.4499 -20.4383 -0.01167 -4.79256 -39.6163 -88.7199 -34.9403 -16.7306 1200 -18.1837 -17.8684 -0.31529 -4.10096 -35.0029 -80.0245 -30.8755 -15.0818 1300 -16.2621 -15.6971 -0.56498 -3.51831 -31.0943 -72.6607 -27.4276 -13.6869 1400 -14.6120 -13.8390 -0.77297 -3.02131 -27.7405 -66.3441 -24.4656 -12.4908 1500 -13.1797 -12.2315 -0.94827 -2.59211 -24.8305 -60.8656 -21.8930 -11.4538 1600 -11.9248 -10.8273 -1.09748 -2.21796 -22.2819 -56.0683 -19.6374 -10.5467 1700 -10.8162 -9.59061 -1.22567 -1.88926 -20.0308 -51.8328 -17.6431 -9.74599 1800 -9.82965 -8.49328 -1.33637 -1.59831 -18.0280 -48.0655 -15.8671 -9.03443 1900 -8.94605 -7.51314 -1.43291 -1.33874 -16.2346 -44.6925 -14.2753 -8.39780 2000 -8.15016 -6.63286 -1.51729 -1.10626 -14.6193 -41.6547 -12.8403 -7.82491 2100 -7.42950 -5.83782 -1.59169 -0.89633 -13.1567 -38.9050 -11.5395 -7.30671 2200 -6.77376 -5.11646 -1.65730 -0.70647 -11.8259 -36.4040 -10.3555 -6.83572 2300 -6.17486 -4.45910 -1.71577 -0.53371 -10.6102 -34.1188 -9.27267 -6.40588 2400 -5.62541 -3.85764 -1.76777 -0.37604 -9.49492 -32.0231 -8.27902 -6.01206 2500 -5.11963 -3.30527 -1.81435 -0.23133 -8.46827 -30.0942 -7.36347 -5.64991 2600 -4.65240 -2.79635 -1.85605 -0.09854 -7.52028 -28.3123 -6.51727 -5.31577 2700 -4.21977 -2.32599 -1.89378 0.02421 -6.64179 -26.6617 -5.73290 -5.00683 2800 -3.81764 -1.89029 -1.92735 0.13752 -5.82584 -25.1284 -5.00380 -4.71991 2900 -3.44314 -1.48515 -1.95798 0.24247 -5.06567 -23.7002 -4.32389 -4.45346 3000 -3.09354 -1.10789 -1.98565 0.34013 -4.35606 -22.3663 -3.68903 -4.20476 3100 -2.76623 -0.75556 -2.01068 0.43094 -3.69183 -21.1177 -3.09427 -3.97213 3200 -2.45918 -0.42577 -2.03341 0.51542 -3.06890 -19.9469 -2.53630 -3.75453 3300 -2.17070 -0.11671 -2.05399 0.59458 -2.48344 -18.8462 -2.01158 -3.55035 3400 -1.89888 0.17365 -2.07253 0.66877 -1.93231 -17.8099 -1.51753 -3.35855 3500 -1.64255 0.44715 -2.08970 0.73821 -1.41260 -16.8321 -1.05114 -3.17771 3600 -1.40046 0.70475 -2.10521 0.80333 -0.92169 -15.9086 -0.61045 -3.00758 3700 -1.17109 0.94806 -2.11914 0.86458 -0.45700 -15.0340 -0.19326 -2.84655 3800 -0.95374 1.17828 -2.13201 0.92234 -0.01681 -14.2051 0.20222 -2.69446 3900 -0.74739 1.39614 -2.14353 0.97654 0.40108 -13.4181 0.57768 -2.55026 4000 -0.55122 1.60273 -2.15396 1.02782 0.79792 -12.6705 0.93446 -2.41338 4100 -0.36458 1.79891 -2.16349 1.07642 1.17577 -11.9585 1.27434 -2.28343 4200 -0.18659 1.98533 -2.17192 1.12215 1.53556 -11.2800 1.59776 -2.16026 4300 -0.01683 2.16295 -2.17978 1.16531 1.87869 -10.6327 1.90671 -2.04271 4400 0.14535 2.33202 -2.18667 1.20636 2.20646 -10.0145 2.20133 -1.93080 4500 0.30061 2.49327 -2.19266 1.24543 2.51949 -9.42327 2.48325 -1.82410 4600 0.44908 2.64745 -2.19836 1.28226 2.81899 -8.85730 2.75276 -1.72216 4700 0.59144 2.79459 -2.20315 1.31716 3.10588 -8.31487 3.01102 -1.62500 4800 0.72782 2.93547 -2.20765 1.35054 3.38088 -7.79469 3.25867 -1.53184 4900 0.85891 3.07036 -2.21145 1.38219 3.64467 -7.29537 3.49632 -1.44281 5000 0.98507 3.19952 -2.21445 1.41209 3.89800 -6.81559 3.72428 -1.35751 5100 1.10617 3.32346 -2.21729 1.44055 4.14127 -6.35429 3.94331 -1.27579 5200 1.22287 3.44252 -2.21965 1.46764 4.37552 -5.91004 4.15411 -1.19747 5300 1.33519 3.55674 -2.22155 1.49363 4.60088 -5.48235 4.35699 -1.12203 5400 1.44355 3.66659 -2.22304 1.51821 4.81794 -5.06998 4.55212 -1.04995 5500 1.54800 3.77219 -2.22419 1.54169 5.02708 -4.67220 4.74049 -0.98025 5600 1.64897 3.87406 -2.22509 1.56403 5.22848 -4.28813 4.92186 -0.91357 5700 1.74651 3.97209 -2.22558 1.58536 5.42330 -3.91735 5.09704 -0.84936 5800 1.84063 4.06644 -2.22581 1.60563 5.61116 -3.55875 5.26626 -0.78746 5900 1.93193 4.15758 -2.22565 1.62499 5.79286 -3.21198 5.42947 -0.72791 6000 2.02009 4.24547 -2.22538 1.64358 5.96843 -2.87631 5.58735 -0.67060

246

. . . .

247

Appendix V

Method for Calculating Chemical Equilibrium of

Combustion Products

This appendix should be read in conjunction with section 3.3. Some inspiration for

the following method is due to C. Olikara and G. L. Borman in the FORTRAN

source code TPEQIL available for download from the University of Wisconsin-

Madison at http://www.erc.wisc.edu/modeling/zerod.php

The moles of atomic carbon, hydrogen, oxygen and nitrogen in the mixture are

determined by summing up the contribution from each species present in the

mixture. The fuel has composition

V-1 Equation set

Five mass balance equations are

where is the total species moles in the mixture.

Eight possible equilibrium equations are listed in Table 3-2 and are repeated here

(note, is in bar)

Appendix V Method for Calculating Chemical Equilibrium of Combustion Products 248

Using six of these equations, six unknown species concentrations can be eliminated

by writing them in terms of and

Further, the carbon mass balance can be combined with the equilibrium equation

to form

and

Then the problem reduces to the solution of four equations in four unknowns

The four equations are:


Total of all fractions

Hydrogen mass balance

Nitrogen mass balance

Oxygen mass balance

To solve this set of equations using the Newton-Raphson method, the partial

derivatives of each function must be found. Referring to the six equilibrium

equations above:

1)

4)

5)

6)

7)

8)

And from the expression for above

The solution of equations f1, f2, f3 and f4 is a challenging numerical problem because

the fractions of the three unknown chemical species and can differ by

many orders of magnitude, as can the equilibrium constants K1-8. Where early

iterations of the Newton-Raphson method resulted in a negative value for one of the

unknown fractions, this was detected and the variable in question was res-set to a

value 100 times less than the previously guessed value. This ensured stability.

Iteration was terminated when all of the primary variables

changed less than 0.1%.


V-2 Low Temperature solution

The lower temperature limit for the full equilibrium calculation was set to 1400K.

Below this temperature, minor species are present in such low fractions (less than 10-

10) as to be negligible. Thus low temperature combustion was assumed with lean side

species being CO2, H2O, N2 and O2 and rich side species being CO, CO2, H2, H2O and

N2. For both situations, moles of nitrogen (diatomic) was found as

For lean mixture the solution in moles is

If the value calculated for is negative, this indicates there is no excess oxygen, so

a rich mixture is in view. The solution is more involved. Setting to zero there

are four unknowns, so the equilibrium equation

is used to close the problem which can be reduced to a single quadratic in

Where

Then


V-3 Equilibrium Initial Approximation

The starting point for an initial guess is the solution for low temperature combustion

converted to mole fractions. Next, for the case of a rich mixture the fraction of

oxygen was estimated by using either

Or

And limiting the result to a maximum value of

Mixtures which were exactly stoichiometric were problematic, so one more step was

taken to remove this problem. The concentrations of the following four species were

updated

This level of initial guess has proved sufficient for evaluating mixtures ranging from

1400K-6000K and equivalence ratio of 0-2.5. The number of iterations in the main

Newton-Raphson routine ranged from 2-10 with an average of about 5.


V-4 Partial derivatives of mixture properties

In some thermodynamic calculations, it is useful to know the rates of change of

mixture properties with respect to one variable. For example, given that for a

mixture of species with mole fractions

Then

Similar expressions can be written for enthalpy .

To evaluate the partials of the gas constant write

Then

Other useful applications for the mixture partial derivatives can be found. For

instance, for a fuel with composition we can write the enthalpy of

combustion at a certain atmospheric condition as


Figure V-1 shows the heat release per kg of n-octane when burned at various

temperatures and equivalence ratios. Numerous interesting features are apparent.

For instance it is obvious that while ever the mixture is lean, more energy can be

released by adding more fuel. However there is a sharp drop in heat release as the

mixture becomes rich of stoichiometric, and the result of adding fuel is an

endothermic reaction. It is also interesting to note that at very high temperatures,

addition of fuel does not release much heat, if any. Note however that if the

combustion products cool, the extra energy will be realised as dissociated species re-

combine.

Figure V-1 Heat release of n-octane at various equivalence ratios and

temperatures

V-4-1 Calculating the partials of each species fraction

To evaluate these mixture property partial derivatives, we need the partials of each

species mole fraction , as well as the total moles . These can be calculated as

follows.

Each of the four mass balance equations can be differentiated with respect to the

variable of interest (either of )

-2.E+7

-1.E+7

0.E+0

1.E+7

2.E+7

3.E+7

4.E+7

5.E+7

0 0.5 1 1.5 2 2.5

1800K

2100K

2400K

2700K 3000K

3300K

3600K

3900K

Equivalence ratio

Mixture made from 75 parts N2, 15 parts O2 by mass and various parts n-octane (C8H18)

3

-dH/

dmfu

el (J

/kg)

1500K

1 bar


Thus

This set of four linear equations can then be solved simultaneously for the four

unknowns .

The value of LHS can be calculated from the dependant species

Note that since the equilibrium constants are functions of temperature (only),

their derivative WRT temperature can be found.

The partial derivatives

are already known from the preceding equilibrium calculation which used the

Newton-Raphson method.

Once the partials of the primary variables

are evaluated, the partials of the remaining species can be evaluated by back

substitution.

255

Appendix VI

Derivation of fundamental one dimensional unsteady

gas equation

This derivation has been adapted from that shown by Blair [24] and Earnshaw [44].

Consider a small element of gas that is impelled by a pressure wave from one

direction as shown in Figure VI-1 (see also Figure 4-2).

Figure VI-1 A fluid element influenced by a pressure wave

Initially, fluid particles are at position , and after some time , they are at position y.

Initially the fluid element has a length . The cross section of the flow remains

constant so the cross section A of the element remains constant. Initially, the

pressure in the infinitesimal element is , and after time the pressure is . The

pressure in element varies according to the volume, and the change in pressure is

assumed isentropic. Thus

(VI-1)

Differentiating this expression WRT gives

(VI-2)

The next relationship to be employed in the solution is the momentum equation:

t0

t

A

x1

Particle path lines

Small fluid element

x2

y2

y1 tim

e

position (x, y)

References 256

For a constant mass this reduces to

which for the fluid element can be written:

(VI-3)

where is the acceleration of a fluid particle or infinitesimal element.

Then (VI-2) can be combined with (VI-3)

(VI-4)

since the speed of sound for a perfect gas is

The integration of equation (VI-4) WRT time will yield the velocity

To integrate this equation, first suppose the solution is of the form:

(VI-5)

Differentiating this equation can WRT gives (after some manipulation)

(VI-6)

This result can be substituted back into equation (VI-4) to give

References 257

Now integrating WRT

where is the constant of integration.

Making two further substitutions using equations (VI-1) and (VI-5) gives

There are two possible results here because we have not had to assume a direction

for the pressure wave.

At the pressure is and the velocity is . Solving the for gives

Thus, the equation for particle velocity is:

(VI-7)

The positive case corresponds to a wave travelling in the positive direction. The

negative case corresponds to a wave moving in the negative direction.

258

. . . .

259

Appendix VII

Derivation of Boundary Flow Equations

The flow at cell boundaries can be calculated as described in section 4.4. Figure

VII-1 shows a typical cell boundary (which could also be a duct boundary). In this

case there are five unknowns – pressure ( ), velocity ( ) and downstream

reference sound speed ( ). However other boundary flows will have various other

combinations of known and unknown quantities. The terms in the equations derived

below will follow the nomenclature of Figure VII-1 where the flow is from 1 2.

Figure VII-1 Schematic of a duct boundary

VII-1 Energy equation

Conservation of energy for a steady flow from 1 2 with gravitational potential

neglected is

Assuming calorically perfect flow this becomes

Position

time

Reflectedwaves Xr

Incident wavesXi

a Ra a0a b Rb

a0b

c Rc a0c

d Rd a0d

a0a a

a0b b

X1 X2 a01 a02

u1 u2

A1 A2

Xia Xib

Unknown values in bold. Flow is from left to right


From the definition of isentropic reference temperature

and writing

and since for a perfect gas

the energy equation becomes

VII-2 Continuity equation

Mass continuity for steady flow from 1 2 is

From the ideal gas relation

Then the mass flow rate is

The continuity equation becomes


VII-3 Wave equation

Gas property discontinuities exist (in general) immediately upstream and

downstream of the boundary flow as sketched Figure VII-1. Since it is a contact

surface between two regions of gas, there is no mass flow across this surface. It

follows therefore, that the contact surface moves with the local fluid and the velocity

of the fluid on either side immediately adjacent to the contact surface is equal.

Further, since the acceleration of the local fluid is finite, as the thickness of the

contact surface approaches zero, the pressure difference across the surface

approaches zero. Thus

And

Writing in terms of the pressure amplitude ratio

Combining the pressure wave equations (4-3) and (4-4) and eliminating one of the

pressure waves we can write for the cell space on each side of the boundary flow

Substituting and for and above gives


VII-4 equation

For thermal energy added to the fluid, the change in temperature (for constant

pressure) is proportional to the energy as

Where is any form of frictional dissipation. Writing

the change in temperature due to friction work is

Writing in terms of isentropic reference temperature

Where is at the pressure at which the thermal energy is being added to the fluid.

263

Appendix VIII

Model Data Structures

This section lists the data structures that were used to form the model. There are

three main categories of object that make an engine model:

Ducts – These are the gas dynamic parts of the model. They are given a length and

cross section, which may be tapered. A duct is made of n cells with n+1 cell

boundaries or nodes. Each end of the duct may be connected to another duct or

volume. If the end is unconnected, the boundary is closed.

Volumes – These are the parts of the model where gas dynamic effects don’t need to

be modelled, such as the atmosphere and cylinder. Volumes are typically larger in

cross section than ducts, and are modelled as zero dimensional thermodynamic

control volumes. A large reservoir such as the atmosphere can be modelled as an

infinitely large volume which supplies a steady pressure to any connecting ducts.

Bodies – the moving parts of the model are represented by bodies. These may be

parts such as pistons and valves. The Pempek free-piston engine had several floating

bodies which interacted through spring forces, contact and friction, such as the

mover and piston mounted passive inlet valves. Bodies can be specified in the

model to constrain the length of certain ducts, determine the volume of certain

volumes (such as the cylinders), or specify the geometric flow area through valves.

Some other data structures control how the above model objects are connected to one

another:

Connect structure – This manages the flow connections between ducts and volumes.

It also stores some flow data at these connections for convenient retrieval.

Body Interaction structure – This manages how the various bodies interact with one

another (if at all). It contains information on the kinds and locations of collisions

possible between each body, the relative distance related forces such as linear spring

force, and relative velocity forces such as kinetic friction.

The code was programmed in Matlab, which implements a convenient data type

“Structure” which uses labelled fields which may contain arrays of differing data

class and size. This is especially useful in the duct structure since the number of

cells may change halfway through a simulation due to automatic re-meshing. Thus

the data structures shown here may need to be modified if coded in a different

language.


Duct data structure

D(d) contains details and flow history for each duct d in the model .name ‘string’ convenient label .description ‘string’ .enabled(t) [Y/N Y/N . . . ducts may be temporarily disabled .friction_function I specifies if and how friction is calculated .heat_transfer_function I specifies if and how heat transfer is calculated .combustion function I specifies if and how combustion is calculated .re_mesh_function I specifies if and how the duct is re-meshed .flow_seperation I specifies if and how flow separation is calculated .A_linear Y/N allows annular ducts to be modelled .mixing 0-1 specifies degree of mixing .conserve_mass Y/N specifies whether mass conservation is enforced .user ? field for customised storage of date .position(1:2) .body I duct end location .offset #.# duct end offset from above location .contributes_to_wall_velocity Y/N

.c(t) the cell based data for the duct at each timestep t .A(c) [# # . . . cross sectional area of each cell .C(c) [# # . . . perimeter of surface of each cell .Cp(c) [# # . . . Specific heat of gas in each cell .Cv(c) [# # . . . Specific heat of gas in each cell

.m(s,c) mass of each species in each cell

.T0(c) [# # . . . reference temperature in each cell Tw(c) [# # . . . wall temperature of each cell

.v_wave(1:2,c) rightward and leftward wave velocity in each cell

.ave_fastest_wave_traverse # duct average of the fastest wave in each cell .dq_dt(c) [# # . . . specific heat transfer rate at each cell .dT0(c) [# # . . . change in ref. temp. due to heat transfer .P_corr(c) [# # . . . pressure correction applied in each cell .m_corr(c) [# # . . . mass correction applied in each cell .dX(c) [# # . . . pressure wave change from heat trans. and P corr .A_end(1:2) [# #] cross sectional area of each end of the duct .C_end(1:2) [# #] perimeter of the surface of each end of the duct

.ket(1:2,c) turbulent KE (for heat transfer model)

.n(t) the node based data for the duct at each timestep t .P(1:2,n) pressure either side of each node

.c(1:2,n) flow velocity either side of each node

.Xi(1:2,n) incident pressure wave on each side of each node

.Xr(1:2,n) reflected pressure wave on each side of each node

.x(n) [# # . . . position of each node .v(n) [# # . . . velocity of each node (if duct is moving)

.dm_dt(s,n) mass flow rate of each species at each node

.T0(1:2,n) reference temp. at each side of each node

.ket(1:2,n) turbulent KE at each side of each node (heat transfer model)


Volume data structure

V(v) contains details and flow history for each volume v in the model .name ‘string’ convenient label .description ‘string’ .enabled(t) [Y/N Y/N . . . volumes may be temporarily disabled .CSA # effective cross s. area. For volume .length details for determining the effective lengh of the vol. .body [# #] up to two bodies to specify effective length of vol. .offset # constant offset to above length .heat_transfer_function I specifies if and how heat transfer is calculated .combustion_function I specifies if and how combustion is calculated .user ? field for customised storage of date .P(t) [# # . . . pressure at each time step .T(t) [# # . . . temperature at each time step .dT_dt(t) [# # . . . temp rate of change at each time step .Cp(t) [# # . . . specific heat at each time step

.Cv(t) [# # . . . specific heat at each time step .Tw(t) [# # . . . wall temp rate of change at each time step .H(t) [# # . . . total enthalpy flow rate at each time step .dq_dt(t) [# # . . . specific heat transfer rate at each time step

.m(s,t) mass of each main species at each time step

.extra_m(s,t) mass of any minor species at each time step

.dm_dt(s,t) mass flow of each main species at each time step.

.dm_ch(s,t) chemical mass rt. of ch. of each species at each time st.

.u(s,t) specific internal energy for each species at each time st.


Body data structure

B(b) contains details and trajectory history for each body b in the model .name ‘string’ convenient label .description ‘string’ .enabled(t) [Y/N Y/N . . . volumes may be temporarily disabled .user ? field for customised storage of date .m # mass of body .x(t) [# # . . . position of body at each timestep .v(t) [# # . . . velocity of body at each timestep .a(t) [# # . . . acceleration of body at each timestep .body_position specifies the method for calculating the body trajectory .index I .Fp specifies any force on the body not due to another body .index I specifies the method for calculating force User_inputs(n) [# # . . . any necessary details .f(t) [# # . . . non-body-body force history

Body Interaction structure

I(b1,b2) contains details and force history for each interacting pair of bodies .upper_collision Y/N .upper_collision_offset # .upper_collision_k 0-1 .lower_collision Y/N .lower_collision_offset # .lower_collision_k 0-1 .Fx(n) specifies up to n different spring type forces between the bodies .index I specifies the force calculation function .user_inputs [# # . . . various necessary details .f # force at current timestep .Fv(n) specifies up to n different friction type forces between the bodies .index I specifies the force calculation function .user_inputs [# # . . . various necessary details .f # force at current timestep


Flow connections structure

C(a,b) contains details and flow history for each flow connection in model .connection Y/N flags the presence of a connection .junction_mode_Cl # specifies the loss coefficient for multi-pipe junctions .volume_velocity # specifies any velocity that would raise the dynamic P .A details for calculating the flow CSA of the connection .body [# #] up to two bodies to specify the flow area .body_offset # offset for the above length .multiplier # multiplier for the opening .Ca I flow area coefficient map to look up .entry Y/N specifies forward or reverse flow .reference # reference area .PR # pressure ratio .flow(t) records history details of the connection flow .c # velocity .P # pressure .T0 # reference temperature .T # temperature .Cp # specific heat .Cv # specific heat

.dm_dt(s) mass flow rate of each species

Flow area coefficient structure

Ca(ca) contains data for specifying the flow area coefficient for certain boundary flows .name ‘string’ convenient label .description ‘string’ .data(1:2) contains the data for forward and reverse flow area coefficients .raw data points from steady flow experiments

.AR [# # . . . area ratio points for fitted data .PR [# # . . . pressure ratio points for fitted data .Ca_fit(PR,AR) net of fited falues corresponding to .AR and .PR

.surf Y/N specifies a surface or line .interp_method ‘string’ method for interpolating Ca_fit ‘linear’ or ‘cubic’

268

. . . .

269

Appendix IX

2nd Order Interpolation – further details

The gas dynamics model described in Chapter 4 is made of a string of idealised

segments (cells) which are constant area, constant property and frictionless. At the

connection point of each of these idealised segments (nodes), any area change, gas

property change or friction is accounted for. Therefore, the pressure and velocity of

the modelled flow usually has some step change across each node, as the effects of

area change, property change and friction modify the flow at this point. The general

situation is illustrated in Figure IX-1, which shows a typical variation in the

rightward pressure wave along a duct at a certain instant in time. The rightward

pressure wave is made up of incident and reflected waves (as described in Figure 4-

9) and the wave’s value is discontinuous at node 2.

Figure IX-1 Interpolating discontinuous pressure waves

The second order interpolation method requires the wave’s value to be known at

three points along the curve. In the example above, the propagation of the rightward

wave is being evaluated – that is, the future value of the wave incident on position

(node) 3 is being evaluated. The values of the three data points are:

The value of the wave at the far ‘upwind’ node (1) is modified for the purpose of the

interpolation calculation to remove the effect of the flow discontinuity at node 2. A

similar procedure is used for the leftward wave.

Rightward wave value

XR

Xr1

position, node

Xr2 Xi2 Xi3

1 2 3


A special problem occurs for the node located second from the end of each duct. In

this case, no third ‘upwind’ node exists (as it lies outside the duct boundary). The

situation is illustrated in Figure IX-2, where the wave incident on node 2 at the

current time step has no third ‘upwind’ node at the previous time step with which to

perform the interpolation.

Figure IX-2 Handling the duct ends

A fake third upwind node (0) is created by using the current time step wave value at

node 1 (which is the duct’s left hand end). This method depends on the duct ends

being evaluated prior to this step, so that legitimate wave values at the duct ends are

already known. The position of the fake upwind node is set to the distance the wave

at this location travels in one time step.

If the duct is very short and only has one cell, then this technique cannot work. This

is because the duct ends are also the problematic ‘second from the end’ nodes. This

is illustrated in Figure IX-3. One of two options may be taken here. The

interpolation may revert to a first order (linear) method with the risk of numerically

unstable extrapolation if the courant number is greater than one, or the fake third

upwind node could be set to the value of the end node (at previous time step), which

should be stable, even at courant numbers greater than one.

Figure IX-3 Single cell ducts

time

previous time-step

Right travelling wave

position, node 1 2 3 4

current time-step

0

time

previous time-step

position, node 1 2

current time-step

271

Appendix X

Re-meshing Criteria and Method

Re-meshing criteria

Automatic mid-simulation duct re-meshing allows individual ducts to maintain

Courant numbers closer to unity. This aim is assisted by the second order wave

interpolation method (see section 4.3.1 and Appendix IX) which allows Courant

numbers somewhat greater than unity.

The process of re-meshing a duct introduces small interpolation errors in all flow and

fluid variables, so the frequency of re-meshing should be minimised. An effective

re-meshing criterion was developed and is described below.

Each cell has a right and left travelling wave which, if there is any local flow

velocity, are propagating at differing speeds. Thus each cell has two Courant

numbers, one for the leftward wave and one for the rightward wave. The higher of

these two Courant numbers are averaged with all the cells in the duct to give the

average Courant number. Also, the highest Courant number for the whole duct is

determined.

The duct is analysed for re-meshing if:

the average Courant number is greater than 1.1 or

the highest Courant number is greater than 1.2 or

the average Courant number is less than 0.85

The optimum number of cells for the duct is then calculated using three methods:

1. the fewest cells which would give an average Courant number greater than 1

2. the most cells which would give an average Courant number less than 1.05

3. the most cell which would give a highest courant number less than 1.15

The result which gives the fewest number of cells (ie the lowest Courant number) is

chosen. Note that the result may be exactly the same as the existing number of cells

in which case there is no change (no-re-meshing) of the duct.

Appendix X Re-meshing Criteria and Method 272

Re-meshing method

In the mesh implementation used here, pressure, velocity, temperature and species

fraction are stored at nodes, while all other duct properties are stored as cell values.

Figure 4-7 is repeated below, showing the nodes as solid circles, and the cell centres

as hollow circles. Note that the cell centres are defined midway between the nodes

both in time and space.

Figure 4-7 Re-meshing a duct

Re-meshing occurs after the most recent cell properties have been evaluated but

before the wave propagation from previous time step to current time step is

calculated. Thus re-meshing involves interpolating nodal values at the previous time

step and cell values that are most recent. Nodes at the current time step are also set

up; however the flow data here are yet to be determined in the calculation sequence

of the model (see section Chapter 6), so no interpolation is necessary.

The nodal and cell data are interpolated using a suitable method, in this case a high

order monotonic fitted curve provided by the MatLab function ‘pchip’. Note that

nodal pressure, velocity and temperature are all defined on both sides of each node,

so there are a total of 6 variables here that need to be interpolated. See Appendix

VIII for a full list of variables for nodes and cells.

time

current time-step

previous time-step

position

273

Appendix XI

Single Shot Experiments Cross Reference

Table XIV-1 lists the corresponding figure numbers in the original publication by

Kirkpatrick [68] for the single shot tests in sections 7.2 to 7.6

Table XIV-1 Cross reference figure numbers for single shot data

Figure number Corresponding figure in [68] Figure number Corresponding

figure in [68] Figure 7-10 Figure 6-14 Figure 7-41 Figure 6-108 Figure 7-11 Figure 6-187 Figure 7-42 Figure 6-168 Figure 7-12 Figure 6-191 Figure 7-43 Figure 6-109 Figure 7-13 Figure 6-18 Figure 7-44 Figure 6-169 Figure 7-14 Figure 6-189 Figure 7-45 Figure 6-56 Figure 7-15 Figure 6-193 Figure 7-46 Figure 6-126 Figure 7-16 Figure 6-26 Figure 7-47 Figure 6-57/127 Figure 7-17 Figure 6-32 Figure 7-48 Figure 6-58 Figure 7-18 Figure 6-7 Figure 7-49 Figure 6-128 Figure 7-19 Figure 6-10 Figure 7-50 Figure 6-59/129 Figure 7-20 Figure 6-162 Figure 7-51 Figure 6-64 Figure 7-21 Figure 6-164 Figure 7-52 Figure 6-134 Figure 7-22 Figure 6-168 Figure 7-53 Figure 6-65/135 Figure 7-23 Figure 6-8 Figure 7-54 Figure 6-66 Figure 7-24 Figure 6-11 Figure 7-55 Figure 6-136 Figure 7-25 Figure 6-15 Figure 7-56 Figure 6-67/138 Figure 7-26 Figure 6-18 Figure 7-57 Figure 6-137 Figure 7-27 Figure 6-175 Figure 7-58 Figure 6-139 Figure 7-28 Figure 6-178 Figure 7-59 Figure 6-140 Figure 7-29 Figure 6-179 Figure 7-60 Figure 6-141 Figure 7-30 Figure 6-180 Figure 7-61 Figure 23 1 Figure 7-31 Figure 6-181 Figure 7-62 Figure 20 1 Figure 7-32 Figure 6-182 Figure 7-63 Figure 6-87 Figure 7-33 Figure 6-183 Figure 7-64 Figure 6-154 Figure 7-34 Figure 6-184 Figure 7-65 Figure 6-88/155 Figure 7-35 Figure 6-185 Figure 7-66 Figure 6-90 Figure 7-37 Figure 6-96 Figure 7-67 Figure 6-156 Figure 7-38 Figure 6-160 Figure 7-68 Figure 6-91/157 Figure 7-39 Figure 6-97 Figure 7-69 Figure 23 [27] Figure 7-40 Figure 6-161 Figure 7-70 Figure 20 [27]

1 This figure from reference [27] Blair, G. P., Kirkpatrick, S. J., Mackey, D. O., and Fleck, R., 1995, Experimental Validation of 1-D Modelling Codes for a Pipe System Containing Area Discontinuities, SAE, Paper 950276

274

. . . .

275

Appendix XII

Derivation of normal shock equations

The derivations presented here are adapted from the derivations presented by

Anderson [13].

Consider the shock shown in Figure XII-1. The shock is perpendicular (normal) to

the direction of the flow. A control volume is constructed around the shock so that

the flow through the control volume is steady.

At the shock there is a change of velocity u, pressure P, temperature T, density and

local speed of sound a. The flow velocity u is defined relative to the shock.

The flow across the shock obeys conservation of energy, mass and momentum. If

the flow is calorically perfect, then an algebraic solution is available.

Figure XII-1 Schematic of a normal shock

The equation for the conservation of energy for a steady, adiabatic flow with no

work, between two states is

If the flow is calorically perfect we can write

and writing

Stationary Shock

Control volume Flow direction



Then

(XII-1)

We now define a state where the flow velocity equals the local sonic velocity

. This is a hypothetical state representing a flow that has undergone some adiabatic

process to accelerate or decelerate it to sonic velocity. The energy equation for this

situation can be expressed as:

Writing the hypothetical sonic flow as a function of the general state and .

(XII-2)

Alternatively:

(XII-3)

Equations (XII-2) and (XII-3) are expressions of the energy equation for a steady

adiabatic flow. Note that no isentropic assumption has been made.

The next step in the solution is to introduce both the conservation of mass and the

conservation of momentum equations, which must hold true for the control volume

of Figure XII-1.

Continuity equation can be written here as:

(XII-4)

The momentum equation is

For the steady flow of Figure XII-1 this can be written


Then the momentum equation can be written as

(XII-5)

Or since


The continuity/momentum equation can be written as:

Substituting equation (XII-3) for and for gives:

This expression can be simplified to:

(XII-6)

With this simple relation we are now able to find the flow on one side of the shock in

terms of known values on the other side. The expressions developed below assume

the upstream state 1 is known, and the downstream state 2 is being calculated, but

the reverse can also be done (that is, calculating the conditions upstream of a shock

based on the downstream conditions).

One further restriction that is not built into the equations here, but is a condition of

second law of thermodynamics, is that the only physically valid shock is one where

the flow is being compressed by the shock. Thus in Figure XII-1 the velocity of the

upstream flow must be greater than the local sonic velocity. The effect of the

shock will then be to slow, compress and heat the flow so that it is sub-sonic on the

downstream side.

Combining equation (XII-6) with equation (XII-2) yields the downstream velocity

as:


(XII-7)

The downstream pressure can now be found by rearranging the momentum equation

(XII-5)

And recalling

so that

(XII-8)

Or substituting away with equation (XII-7)

(XII-9)

Finding the remaining downstream values is straight forward, for instance from the

ideal gas relation

Using the continuity equation (XII-4)

(XII-10)

Then

(XII-11)

And


(XII-12)

Alternatively, if the speed of the shock is unknown but the pressure ratio across the

shock is known, the shock speed can be calculated by re-arranging equation (XII-

9)

(XII-13)

The downstream velocity can now be found by employing equation (XII-7)

(XII-7)

or by writing equation (XII-8) as

(XII-14)

The change in velocity across the shock in terms of the pressure ratio can be found

by writing equation (XII-14) as

And substituting in equation (XII-13)

(XII-15)

The normal shock equations derived here can be used for the case of a moving shock

by examining Figure XII-2 and noting that the shock velocity relative to the pre-

shocked gas is

and


Then the velocity of the post shock gas v is

v v

v

Figure XII-2 Travelling shock

Pre shock Post shock

+ve direction Travelling Shock

281

Appendix XIII

Rayleigh and Fanno Flow

XIII-1 One dimensional flow with heat transfer (Rayleigh

flow)

The momentum equation for frictionless flow in a constant area duct is given by

equation (XII-5)

Since according to the continuity equation

and for a perfect gas

The momentum equation can be written

It is useful to use the Mach number, defined as

Thus

(XIII-1)

Or alternatively simply

or

From the ideal gas equation and the continuity equation


From the definition of Mach number and since

thus

(XIII-2)

And again

(XIII-3)

For a flow from the total heat per kilogram added to the flow can be found by

applying the energy equation (XII-1)

or

(XIII-4)

If heat transfer is specified in the problem, along with flow conditions at a certain

point, then the solution can be found by plotting the flow as a function of Mach

number or velocity using equations (XIII-1) to (XIII-4), and locating the Mach

number where heat transfer is equal to that specified. This is a trial and error

approach. Note that the maximum heat that can be added to a flow will be when the

resulting flow is sonic.


XIII-2 One dimensional flow with friction (Fanno flow)

The energy equation for steady adiabatic flow of a perfect gas equation (XII-1) is

The Mach number is defined as

Thus

(XIII-5)

Or alternatively simply

From the ideal gas equation and the continuity equation

From the definition of Mach number and since

Then

(XIII-6)

And again


(XIII-7)

For a flow from with friction the total shear force on the wall can be found by

applying the momentum equation

For constant area flow with friction this becomes

Where is the friction force and is directed opposite to the flow.

Thus

And noting

then

(XIII-8)

If friction force is specified in the problem, along with flow conditions at a certain

point, then the solution can be found by plotting the flow as a function of Mach

number or velocity using equations (XIII-5) to (XIII-8), and locating the Mach

number where friction is equal to that specified. Since friction force is generally a

function of flow velocity an iterative approach may be needed. Note that the

maximum friction force that can be imposed on a flow will be when the resulting

flow is sonic. Thus it is a physical impossibility for a flow that is subsonic to driven

supersonic by friction alone, and likewise for an initially supersonic flow to be

driven subsonic by friction alone.

285

Appendix XIV

Graphical user interface screen shots

To assist in creating engine models, a graphical user interface was created. This

allowed individual engine model creation to be carried out quickly and accurately. It

also facilitated rapid and accurate editing of models to test the results of design

changes. A further extension of the graphical user interface was for post processing

the simulation results. This was necessary given the large volume of time-series data

contained in a single simulation.

Figure XIV-1 shows the main screen which is used to load saved models, edit basic

simulation parameters (such as time step number and size), and launch the model

editing screens. Figure XIV-2 shows the Duct creation/editing screen, where the

parameters of each duct in the model are set. Figure XIV-3 and likewise shows the

Volume editing screen. Connections between ducts to ducts, volumes to volumes

and volumes to ducts are also specified here and can be set from either screen.

Figure XIV-4 shows the Body editing screen. This screen also allows the user to

specify the relationships between bodies (if any) such as collisions, friction, spring

forces and user customisable forces (such as generator force). Figure XIV-5 shows

the Flow Coefficients screen. This allows the user to add and edit flow area

coefficients for various situations. This database also contains valve aerodynamic

force coefficients that are used to calculate the dynamics of the inlet and exhaust

valves. Figure XIV-6 shows the Function editing screen. This allows the user to add

and edit the arguments and return variables for a range of customisable functions.

Figure XIV-7 shows the screen for setting up animation plots. Six example

animations are included in the data CD that is packaged with this thesis, and are

listed in Appendix XV. Figure XIV-8 shows and example animation screen grab.

The data series at any time step can be exported to the clip-board. Figure XIV-9

shows the time-series plotter. This screen allows any time series data from the

model to be plotted, and optionally exported to the clip-board.


Figure XIV-1 Main screen

Figure XIV-2 Edit Ducts screen


Figure XIV-3 Edit Volumes screen

Figure XIV-4 Edit Bodies screen


Figure XIV-5 Edit Area Coefficients screen

Figure XIV-6 Edit Functions screen


Figure XIV-7 Create Animated plot screen

Figure XIV-8 Example Animation screen grab (shock tube problem)


Figure XIV-9 History plot screen

291

Appendix XV

Table of contents of data CD

Thesis.pdf Thesis soft copy

points.mp4 3CFD scavenging animation

existing engine motored.mp4 1D Motored engine simulation (see section 7.7)

existing engine fired.mp4 1D Fired engine simulation (see section 7.7)

modified engine 22mg.mp4 1D Fired simulation of modified passive inlet valve

layout with lower compressor pressure and tuned

exhaust pipe (see section 8.1)

port scavenged 2_5mg.mp4 1D port scavenged simulation, 2.5mg fuel injection

(see section 8.3)

port scavenged 5mg.mp4 1D port scavenged simulation, 5mg fuel injection

(see section 8.3)

port scavenged 22_5mg.mp4 1D port scavenged simulation, 22.5mg fuel

injection (see section 8.3)

numerical modelling of the gas dynamics of a prototype free-piston engine · pdf...

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